Definite Integration

Introduction to Definite Integrals

Up to now, we’ve studied the Indefinite Integral, which is just the function that you get when you integrate another function.

The definite integral is actually a number that represents the area under the curve of that function (above the $ x$-axis) from an “$ x$” position to another “$ x$” position; we learned how to get this area using Riemann Sums. (And don’t forget that with non-curved figures, we can get the area under a curve without using Calculus, but just using Geometry!)

It’s not obvious that an integral is an area under a curve, but it helps when you think of the equation $ \text{Distance}=\text{Rate}\,(\text{Velocity})\times \text{Time}$: for the case when the $ x$-axis of the curve represents time, and $ y$-axis represents rate, the area (length times width) can represent a distance (or change in position). We can use this principle to determine how much something changes (for example, its distance) over time.

Here’s the formal Definite Integral as the Area of a Region:

Definite Integral as the Area of a Region

Let $ f$ be continuous and above the $ y$-axis (non-negative) on interval $ [a,b]$. The area of the region bounded by $ f$, the $ x$-axis and vertical lines at $ x=a$ and $ x=b$ (lower and upper limits) is:

$ \displaystyle \text{Area}=\int\limits_{a}^{b}{{f\left( x \right)}}\,dx$

We’ll see soon that to get this area, we take the integral of $ f\left( x \right)$, plug $ b$ in for $ x$ and then subtract from that value what we get by plugging in $ a$ for $ x$.

 

Note: The area of the region represented by an integral is only applicable if the region in the interval is totally above the $ \boldsymbol{x}$-axis (positive $ y$). We’ll learn later that if any part of the graph is below the $ x$-axis (negative $ y$), to get that definite integral, we’ll take the “negative of the area”. Thus, to get the definite integral of a function that is both above and below the $ x$-axis, we can subtract the area above the $ x$-axis by the area below the $ x$-axis in that interval.

Properties of Definite Integrals

Definite Integrals have some properties; think of these properties just like the properties of any type of area. Most are somewhat obvious:

Properties of Definite Integrals

  • $ \displaystyle \int\limits_{a}^{a}{{f\left( x \right)}}\,dx=0$, if the function is defined at $ x=a$ (If you stay at one point, you don’t have any area.)
  • $ \displaystyle \int\limits_{b}^{a}{{f\left( x \right)}}\,dx=-\int\limits_{a}^{b}{{f\left( x \right)}}\,dx$, if the function is integrable on $ [a,b]$. (Think of going backwards and “erasing” area.)
  • $ \displaystyle \int\limits_{c}^{a}{{f\left( x \right)}}\,dx=\int\limits_{a}^{b}{{f\left( x \right)}}\,dx+\int\limits_{b}^{c}{{f\left( x \right)}}\,dx$, if the function is integrable on $ [a,c]$. (You can add areas; usually this applies when $ b$ is between $ a$ and $ c$, but it always works if $ f\left( x \right)$ is integrable on the largest interval.)
  • $ \displaystyle \int\limits_{a}^{b}{{k\cdot f\left( x \right)}}\,dx=k\cdot \int\limits_{a}^{b}{{f\left( x \right)}}\,dx$, if the function is integrable on $ [a,b]$. (You can move scalars to outside of areas/integrals.)
  • $ \displaystyle \int\limits_{a}^{b}{{f\left( x \right)}}\,\pm g\left( x \right)dx=\int\limits_{a}^{b}{{f\left( x \right)}}\,dx\,\,\pm \,\,\int\limits_{a}^{b}{{g\left( x \right)}}\,dx$, if the function is integrable on $ [a,b]$. (You can split up areas/integrals).

Now let’s do some problems that demonstrate the definite integral as an area:

Definite Integral Problem Solution           Definite Integral Problem Solution
Set up a definite integral that yields the following area:

$ f\left( x \right)=4$

$ \displaystyle f\left( x \right)=\int\limits_{0}^{3}{{4\,dx}}$

 

(We can see that the area is $ b\cdot h=\left( 3 \right)\left( 4 \right)=12$ )

Set up a definite integral that yields the following area:

$ f\left( x \right)={{x}^{2}}+3$

$ \displaystyle f\left( x \right)=\int\limits_{{.5}}^{2}{{\left( {{{x}^{2}}+3} \right)dx}}$

 

(We can’t easily get the area geometrically)

Sketch a graph whose area is given by the definite integral.

 

Then use a geometric formula to find evaluate the integral.

$ \displaystyle \int\limits_{0}^{3}{{x\,dx}}$

Area (Triangle) =

$ \displaystyle \frac{1}{2}bh=\frac{1}{2}\left( 3 \right)\left( 3 \right)=4.5$

Sketch a graph whose area is given by the definite integral.

 

Then use a geometric formula to find evaluate the integral.

$ \displaystyle \int\limits_{{-3}}^{3}{{\left( {3-\left| x \right|} \right)dx}}$

Area (Triangle) =

$ \displaystyle \frac{1}{2}bh=\frac{1}{2}\left( 6 \right)\left( 3 \right)=9$

Here are a few problems that illustrate the properties of definite integrals. Note that not all of these integrals may be areas, since some are negative; we’ll soon learn that if part of the function is under the $ \boldsymbol{x}$-axis, the integral is a “negative area”.

Definite Integral Problem Solution      
Evaluate the integrals, given the following values:

$ \displaystyle \int\limits_{3}^{5}{{dx}}=4\,\,\,\,\,\,\,\,\,\int\limits_{3}^{5}{{x\,dx}}=10\,\,\,\,\,\,\,\,\,\int\limits_{3}^{5}{{{{x}^{2}}dx}}=50$

 

  a. $ \displaystyle \int\limits_{5}^{3}{{x\,dx}}$       b. $ \displaystyle \int\limits_{3}^{5}{{4x\,dx}}$      c. $ \displaystyle \int\limits_{3}^{5}{{\left( {\frac{1}{2}{{x}^{2}}+3x-8} \right)dx}}$

a. $ \displaystyle \int\limits_{5}^{3}{{x\,dx}}=-\int\limits_{3}^{5}{{x\,dx}}=-10$

b. $ \displaystyle \int\limits_{3}^{5}{{4x\,dx}}=4\int\limits_{3}^{5}{{x\,dx}}=\left( 4 \right)\left( {10} \right)=40$

c. $ \displaystyle \begin{align}\int\limits_{3}^{5}{{\left( {\frac{1}{2}{{x}^{2}}+3x-8} \right)dx}}&=\frac{1}{2}\int\limits_{3}^{5}{{{{x}^{2}}dx+3\int\limits_{3}^{5}{{x\,dx-8\int\limits_{3}^{5}{d}}}x}}\\&=\frac{1}{2}\left( {50} \right)+3\left( {10} \right)-8\left( 4 \right)=23\end{align}$

Evaluate the integrals, given the following values:

$ \displaystyle \int\limits_{1}^{4}{{f\left( x \right)dx}}=5\,\,\,\,\,\,\,\,\int\limits_{4}^{8}{{f\left( x \right)dx}}=-3\,\,\,\,\,\,\,\,\int\limits_{4}^{8}{{g\left( x \right)dx}}=15$ 

 

  a. $ \displaystyle \int\limits_{1}^{8}{{f\left( x \right)dx}}$       b. $ \displaystyle \int\limits_{4}^{1}{{-4f\left( x \right)dx}}$   

c. $ \displaystyle \int\limits_{4}^{8}{{\left[ {3f\left( x \right)-g\left( x \right)} \right]\,dx}}$       d. $ \displaystyle \int\limits_{4}^{4}{{4f\left( x \right)dx}}$

a. $ \displaystyle \int\limits_{1}^{8}{{f\left( x \right)dx}}=\int\limits_{1}^{4}{{f\left( x \right)dx}}+\int\limits_{4}^{8}{{f\left( x \right)dx}}=5+-3=2$

b. $ \displaystyle \begin{align}\int\limits_{4}^{1}{{-4f\left( x \right)dx}}&=-4\int\limits_{4}^{1}{{f\left( x \right)x\,dx}}=-4\left( {-\int\limits_{1}^{4}{{f\left( x \right)x\,dx}}} \right)\\\,\,&=\left( {-4} \right)\left( {-5} \right)=20\end{align}$

c. $ \displaystyle \begin{align}\int\limits_{4}^{8}{{\left[ {3f\left( x \right)-g\left( x \right)} \right]\,dx}}&=3\int\limits_{4}^{8}{{f\left( x \right)-\int\limits_{4}^{8}{{g\left( x \right)}}}}\\\,\,&=3\left( {-3} \right)-15=-24\end{align}$

d.  $ \displaystyle \int\limits_{4}^{4}{{4f\left( x \right)dx}}=4\int\limits_{4}^{4}{{f\left( x \right)dx}}=4\left( 0 \right)=0$

Fundamental Theorem of Calculus

Wow! This sounds important, doesn’t it? That’s because the Fundamental Theorem of Calculus is important; this theorem is used around the world every day to obtain areas (among other things) of all sort of objects. And the great thing about this theorem is it’s so simple to use, especially compared to some of the summing techniques we’ve used. This theorem is also called the “Net Change” Theorem. Here goes:

Fundamental Theorem of Calculus

If a function $ f\left( x \right)$ is continuous on a closed interval $ [a,b]$, and $ F$ is an indefinite integral (antiderivative) on that same interval, then:

$ \displaystyle \int\limits_{a}^{b}{{f\left( x \right)}}\,dx=F\left( b \right)-F\left( a \right)$

For example, to evaluate $ \displaystyle \int\limits_{1}^{2}{{{{x}^{4}}dx}}$:            $ \displaystyle \int\limits_{1}^{2}{{{{x}^{4}}}}\,dx=\left[ {\frac{{{{x}^{5}}}}{5}} \right]_{1}^{2}=\frac{{{{{\left( 2 \right)}}^{5}}}}{5}-\frac{{{{{\left( 1 \right)}}^{5}}}}{5}=\frac{{32}}{5}-\frac{1}{5}=\frac{{31}}{5}$

Note that if the function is totally above the $ y$-axis in the given interval, this calculation is the area between the function and the $ x$-axis. If the function is totally below the $ x$-axis in this interval, this calculation is the opposite (negative) of the area between the $ x$-axis and the function. If this function is both above and below the $ x$-axis, then this calculation is the area above the $ x$-axis subtracted by the area below the $ x$-axis.

Definite Integrals on the Graphing Calculator

You can evaluate definite integrals in the graphing calculator using the math fnInt(, much like you used the nDeriv( for derivatives.

Hit MATH and then scroll down to fnInt( (or hit 9). Put the lower and upper values for the interval and type in the function using the X,T,θ,n key, hitting the right arrow key in between each entry. Then put “$ x$” after the “$ d$” for “$ dx$”, using right arrow key again. (In the case of $ \displaystyle \int\limits_{1}^{2}{{{{x}^{4}}}}\,dx$, we had to use the arrow key twice, since we had to use it after the exponent to go back down.) You can also go back with the left arrow key if you need to make any changes. Then hit “Enter”:

Note that if you have graphed a function in Y=, you can also use 2nd trace (calc) 7 ($ \displaystyle \int{{f\left( x \right)dx}}$) to integrate the function, and you will be asked to enter lower and upper limits.

Here are some  Definite Integration problems. Notice that when we are taking the definite integral of an Absolute Value function, we need to split the function at the points where the absolute values equals 0, and then, as we did in the Piecewise Functions section, either use the original function, or negate the function, depending on the sign of the function (without the absolute value) in that interval. Note that you can check these using fnInt (MATH 9) on your graphing calculator.

Definite Integral Problem

Solution          

 $ \displaystyle \int\limits_{1}^{3}{{5x\,dx}}$

$ \displaystyle \int\limits_{1}^{3}{{5x\,dx}}=\left[ {\frac{{5{{x}^{2}}}}{2}} \right]_{1}^{3}=\frac{{5{{{\left( 3 \right)}}^{2}}}}{2}-\frac{{5{{{\left( 1 \right)}}^{2}}}}{2}=\frac{{45}}{2}-\frac{5}{2}=20$

 $ \displaystyle \int\limits_{0}^{8}{{\left( {\sqrt[3]{t}-4} \right)dx}}$ $ \displaystyle \begin{align}\int\limits_{0}^{8}{{\left( {{{t}^{{\frac{1}{3}}}}-4} \right)dt}}&=\left[ {\frac{{{{t}^{{\frac{4}{3}}}}}}{{\frac{4}{3}}}-4t} \right]_{0}^{8}=\left[ {\frac{{3{{t}^{{\frac{4}{3}}}}}}{4}-4t} \right]_{0}^{8}\\&=\left( {\frac{{3{{{\left( 8 \right)}}^{{\frac{4}{3}}}}}}{4}-4\left( 8 \right)} \right)-\left( {\frac{{3{{{\left( 0 \right)}}^{{\frac{4}{3}}}}}}{4}-4\left( 0 \right)} \right)=-20\end{align}$
 $ \displaystyle \int\limits_{0}^{{\frac{\pi }{2}}}{{\frac{{\cot x-\cos x}}{{\cot x}}\,dx}}$ $ \displaystyle \begin{align}\int\limits_{0}^{{\frac{\pi }{2}}}{{\frac{{\cot x-\cos x}}{{\cot x}}\,dx}}&=\int\limits_{0}^{{\frac{\pi }{2}}}{{1-\frac{{\cos x}}{{\left( {\frac{{\cos x}}{{\sin x}}} \right)}}\,=\int\limits_{0}^{{\frac{\pi }{2}}}{{1-\sin x\,dx}}=\left[ {x+\cos x} \right]_{0}^{{\frac{\pi }{2}}}dx}}\\\,\,\,\,\,&=\left[ {\frac{\pi }{2}+\cos \left( {\frac{\pi }{2}} \right)} \right]-\left[ {0+\cos \left( 0 \right)} \right]=\left( {\frac{\pi }{2}+0} \right)-\left( {0+1} \right)\&=\frac{\pi }{2}-1\approx .571\,\,\,\,\,\end{align}$

$ \displaystyle \int\limits_{{-5}}^{5}{{\left| {{{x}^{2}}-4} \right|\,dx}}$

The graph:

Note that we can integrate with a calculator:

First, create a Sign Chart to see where the function is positive or negative:

For the positive intervals, use $ {{x}^{2}}-4$ as is. Where it is negative, negate $ {{x}^{2}}-4$ to get $ 4-{{x}^{2}}$. This is because the absolute value function only returns a positive function, so when we remove the absolute value to integrate, we need to adjust the underlying function. Separate into three integrals for these intervals:

$ \displaystyle \begin{align}\int\limits_{{-5}}^{5}{{\left| {{{x}^{2}}-4} \right|\,dx}}&=\int\limits_{{-5}}^{{-2}}{{\left( {{{x}^{2}}-4} \right)dx+}}\int\limits_{{-2}}^{2}{{\left( {-{{x}^{2}}+4} \right)dx}}+\int\limits_{2}^{5}{{\left( {{{x}^{2}}-4} \right)dx}}\\&=\left[ {\left( {\frac{{{{x}^{3}}}}{3}-4x} \right)} \right]_{{-5}}^{{-2}}+\left[ {\left( {-\frac{{{{x}^{3}}}}{3}+4x} \right)} \right]_{{-2}}^{2}+\left[ {\left( {\frac{{{{x}^{3}}}}{3}-4x} \right)} \right]_{2}^{5}\\\,&=\left[ {\left( {\frac{{{{{\left( {-2} \right)}}^{3}}}}{3}-4\left( {-2} \right)} \right)-\left( {\frac{{{{{\left( {-5} \right)}}^{3}}}}{3}-4\left( {-5} \right)} \right)} \right]+\left[ {\left( {-\frac{{{{{\left( 2 \right)}}^{3}}}}{3}+4\left( 2 \right)} \right)-\left( {-\frac{{{{{\left( {-2} \right)}}^{3}}}}{3}+4\left( {-2} \right)} \right)} \right]\\&\,\,\,\,\,\,+\left[ {\left( {\frac{{{{{\left( 5 \right)}}^{3}}}}{3}-4\left( 5 \right)} \right)-\left( {\frac{{{{{\left( 2 \right)}}^{3}}}}{3}-4\left( 2 \right)} \right)} \right]=27+\frac{{32}}{3}+27=\frac{{194}}{3}=64\frac{2}{3}\end{align}$

And, again, we can use the definite integral to get an area, if the $ y$-values in the interval are greater than 0 (the function is completely above the $ x$-axis). Note in the second problem, we have to solve for the $ x$-intercepts, or zeros, and sketch a graph (or use a Sign Chart) to see where the function lies above the $ x$-axis.

Problem Solution  
Find the area of the region with the following boundaries:

$ f\left( x \right)=3x-5,\,\,y=0,\,\,x=2,\,\,x=4$

Since this region is all above the $ \boldsymbol{x}$-axis, we can use the definite integral to get the area:

$ \displaystyle \begin{align}\int\limits_{2}^{4}{{\left( {3x-5} \right)dx}}&=\left[ {\frac{3}{2}{{x}^{2}}-5x} \right]_{2}^{4}\\&=\left[ {\frac{3}{2}{{{\left( 4 \right)}}^{2}}-5\left( 4 \right)} \right]-\left[ {\frac{3}{2}{{{\left( 2 \right)}}^{2}}-5\left( 2 \right)} \right]\\&=4-\left( {-4} \right)=8\end{align}$

Find the area of the region with the following boundaries:

$ f\left( x \right)=3\sqrt{x}-x,\,\,\,y=0$

First, find the zeros to get the interval where the function is above the $ x$-axis:

$ \begin{array}{l}3\sqrt{x}-x=0;\,\,\,\,\,\,3\sqrt{x}=x;\,\,\,\,\,{{\left( {3\sqrt{x}} \right)}^{2}}={{x}^{2}};\,\,\,\,\,\,\,9x={{x}^{2}};\\\,\,\,\,\,{{x}^{2}}-9x=0;\,\,\,\,x\left( {x-9} \right)=0;\,\,\,\,\,\,\,x=0\text{ and }x=9\end{array}$

 

Thus, we need $ \displaystyle \int\limits_{0}^{9}{{\left( {3\sqrt{x}-x} \right)dx}}$:        $ \displaystyle \begin{align}\int\limits_{0}^{9}{{\left( {3\sqrt{x}-x} \right)dx}}&=\int\limits_{0}^{9}{{\left( {3{{x}^{{\frac{1}{2}}}}-x} \right)dx}}=\left[ {3\cdot \frac{{{{x}^{{\frac{3}{2}}}}}}{{\left( {\frac{3}{2}} \right)}}-\frac{{{{x}^{2}}}}{2}} \right]_{0}^{9}\\&=\left[ {3\cdot \frac{2}{3}{{{\left( 9 \right)}}^{{\frac{3}{2}}}}-\frac{{{{9}^{2}}}}{2}} \right]-\left[ {3\cdot \frac{3}{2}{{{\left( 0 \right)}}^{{\frac{3}{2}}}}-\frac{{{{0}^{2}}}}{2}} \right]=\left( {54-\frac{{81}}{2}} \right)-0=13.5\end{align}$

Definite Integration and Area

You’ve probably realized by now (and I’ve hinted at it a few times) that to get the value of an integral under the $ x$-axis, you can take the opposite (negative) of the area of that region. Thus, you have to really be careful if a problem calls for an integral of a region that is both above and below the $ x$-axis, you have to basically add up the area above the $ x$-axis and subtract the area below the $ x$-axis.

But let’s think about it. If you were to take the absolute value of the function (so that everything moves above the $ x$-axis), you would have the area! Here’s an example:

Definite Integrals and Area

Problem: Below is the graph of a function $ f$, which consists of straight lines, a semi-circle, and a quarter circle.

Evaluate:   a. $ \displaystyle \int\limits_{{-8}}^{9}{{f\left( x \right)dx}}$      b. $ \displaystyle \int\limits_{{9}}^{5}{{f\left( x \right)dx}}$      c. $ \displaystyle \int\limits_{{-8}}^{9}{{\left| {f\left( x \right)} \right|dx}}$      d. $ \displaystyle \int\limits_{0}^{3}{{\left( {f\left( x \right)+2} \right)dx}}$      e. $ \displaystyle \int\limits_{2}^{4}{{f\left( {x-1} \right)dx}}$      f. $ \displaystyle \int\limits_{{-4}}^{4}{{f\left( {\left| x \right|} \right)dx}}$     g. If $ F\left( {-8} \right)=6$ and $ F$ is the antiderivative of $ f$, find $ F\left( 9 \right)$.

Solution: Use Geometry to figure out the area in each section; the graph above is annotated using rectangle, triangle, and circle area formulas.

a. $ \displaystyle \int\limits_{{-8}}^{9}{{f\left( x \right)dx}}$:    Add up the areas in all the 9 regions, using negatives for the areas below the $ x$-axis: $ -2+2+8+6.28+6+2+\left( {-.5} \right)+\left( {-3} \right)+\left( {-7.07} \right)=11.71$.

b. $ \displaystyle \int\limits_{9}^{5}{{f\left( x \right)dx}}$:    Add up the areas between $ x=5$ and $ x=9$, but since we’re going “backwards” (from 9 to 5), take the opposite (negative): $ \left( {-.5} \right)+\left( {-3} \right)+\left( {-7.07} \right)=-10.57\,\,\,\,\,\,\,\,\,-\left( {-10.57} \right)=10.57$.

c. $ \displaystyle \int\limits_{{-8}}^{9}{{\left| {f\left( x \right)} \right|dx}}$:     Add up the absolute value of the areas in all the 9 regions (so disregard the negatives): $ 2+2+8+6.28+6+2+.5+3+7.07=36.85$.

d.  $ \displaystyle \int\limits_{0}^{3}{{\left( {f\left( x \right)+2} \right)dx}}$ (Vertical Shift Up):    Take the area between $ x=0$ and $ x=3$ (area is 6), and then imagine the top of this rectangle (the $ y$) is translated up 2 to 4. The new area is $ 6+6=12$.

e. $ \displaystyle \int\limits_{2}^{4}{{f\left( {x-1} \right)dx}}$  (Horizontal Shift to Right):    The whole graph is shifted to the right by 1, so to get back to the original graph, subtract 1 from the upper and lower bounds:  $ \displaystyle \int\limits_{2}^{4}{{f\left( {x-1} \right)dx}}=\int\limits_{1}^{3}{{f\left( x \right)dx}}=4$. See how $ \displaystyle f\left( {x-1} \right)$ at $ x=4$ is the same as $ \displaystyle f\left( {x} \right)$ at $ x=3$?

f.  $ \displaystyle \int\limits_{{-4}}^{4}{{f\left( {\left| x \right|} \right)dx}}$ (Absolute value of the $ x$):   “Erase” what’s to the left of the $ y$-axis (the negative $ x$-values) and reflect what’s on the right (positive $ x$-values from $ x=0$ to $ x=4$), so the function is even. Double the area of the positive $ x$ (which is all above the $ x$-axis), so we have: $ 2\left( {6+1.5} \right)=15$.

g.  This one’s a little tricky; use the equation $ \displaystyle \int\limits_{{-8}}^{9}{{f\left( x \right)dx}}=F\left( 9 \right)-F\left( {-8} \right)$.  Thus, $ \displaystyle F\left( 9 \right)=\int\limits_{{-8}}^{9}{{f\left( x \right)dx}}+F\left( {-8} \right)=11.71+6=17.71$.

Here are a few more problems on Definite Integration and Area:

Definite Integral Properties

Problems: Supposed that $ f$ and $ g$ are continuous functions defined on the interval $ [1, 4]$, such that: $ \displaystyle \int\limits_{1}^{4}{{f\left( x \right)dx=5}}$ and $ \displaystyle \int\limits_{1}^{4}{{g\left( x \right)dx=-3}}$

Evaluate:   a. $ \displaystyle \int\limits_{1}^{4}{{h\left( x \right)dx=5}}$, where $ h\left( x \right)=f\left( x \right)+2$        b. $ \displaystyle \int\limits_{{-1}}^{2}{{k\left( x \right)dx}}$, where $ k\left( x \right)=f\left( {x+2} \right)$        c. $ \displaystyle \int\limits_{2}^{5}{{j\left( x \right)dx}}$, where $ j\left( x \right)=4g\left( x \right)$        d. $ \displaystyle \int\limits_{1}^{4}{{\left( {2f-g} \right)dx}}$

Solutions: 

a. (Vertical Shift Up) Since $ \displaystyle \int\limits_{1}^{4}{{f\left( x \right)dx=5}}$ and we are shifting the function vertically up 2, add another “block” of area that is 2 units high by 3 units wide. The new area is $ 5+2\cdot 3=11$ (see diagram). This makes sense since $ \displaystyle \int\limits_{1}^{4}{{\left( {f\left( x \right)+2} \right)}}=\int\limits_{1}^{4}{{f\left( x \right)}}+\int\limits_{1}^{4}{2}=\int\limits_{1}^{4}{{f\left( x \right)}}+\left. {2x} \right|_{1}^{4}=5+\left( {8-2} \right)=11$.         

b. (Horizontal Shift to Left) For $ f\left( {x+2} \right)$, the function $ f\left( x \right)$ is shifted to the left by 2, so for $ \displaystyle  \int\limits_{{-1}}^{2}{{f\left( {x+2} \right)dx}}$, add 2 to the upper and lower limits and use $ f\left( x \right)$. Thus, $ \displaystyle  \int\limits_{{-1}}^{2}{{f\left( {x+2} \right)dx}}=\int\limits_{1}^{4}{{f\left( x \right)dx}}=5$.

c. Since we have a scalar out in front of the $ g\left( x \right)$, just multiply the integral amount by that scalar: $ \displaystyle \int\limits_{1}^{4}{{\left[ {4g\left( x \right)} \right]dx=4\int\limits_{1}^{4}{{g\left( x \right)dx=4\cdot -3}}}}=-12$.

d. We can add and subtract integrals, as well as multiply them by scalars: $ \displaystyle \int\limits_{1}^{4}{{\left( {2f-g} \right)dx}}=\int\limits_{1}^{4}{{\left[ {2f\left( x \right)} \right]dx-\int\limits_{1}^{4}{{g\left( x \right)dx=2\int\limits_{1}^{4}{{f\left( x \right)dx-\int\limits_{1}^{4}{{g\left( x \right)dx=2\cdot 5-}}}}\left( {-3} \right)}}}}=13$.

Definite Integral Properties Problems

Problems: Find $ \displaystyle \int\limits_{2}^{6}{{f\left( x \right)dx}}$, if:      a. $ f\left( x \right)$ is even, $ \displaystyle \int\limits_{{-2}}^{2}{{f\left( x \right)dx}}=8$ and $ \displaystyle \int\limits_{{-6}}^{6}{{f\left( x \right)dx}}=20$        b. $ f\left( x \right)$ is odd, and $ \displaystyle \int\limits_{{-2}}^{6}{{f\left( x \right)dx=8}}$       c. $ \displaystyle \int\limits_{2}^{6}{{\left( {2f\left( x \right)-1} \right)dx}}=14$

Solutions: 

a. The best way to do these types of problems is to draw a picture. An even function is symmetrical around the $ y$-axis, meaning any area on the right-hand side (positive $ x$) is the same as that on the left-hand side (negative $ x$), for the same distance from the origin. Divide up the areas given ($ 4+4=8$ and $ 10+10=20$) to get the area in the interval $ [2,6]$ to be $ 6$.         

b. Again, draw a picture. Since the function is odd, it’s symmetrical around the origin, meaning that any area on the right-hand side (positive $ x$) is the opposite (negative) of that on the left-hand side (negative $ x$), for the same distance from the origin. Thus, the areas in the interval $ [-2,0]$ and $ [0,2]$ cancel out (add up to 0). This makes the area in the interval $ [2,6]$ the same as the area in the interval $ [-2,6]$, which is $ 8$.           

c. Break up the integral to solve back for $ \displaystyle \int\limits_{2}^{6}{{f\left( x \right)dx}}$:

$ \displaystyle \begin{align}14&=\int\limits_{2}^{6}{{\left( {2f\left( x \right)-1} \right)dx}}=2\int\limits_{2}^{6}{{f\left( x \right)dx-\int\limits_{2}^{6}{{1dx}}}}=2\int\limits_{2}^{6}{{f\left( x \right)dx-\left. x \right|_{{x=2}}^{{x=6}}}}=2\int\limits_{2}^{6}{{f\left( x \right)dx-\left( {6-2} \right)}}\\14&=2\int\limits_{2}^{6}{{f\left( x \right)dx-4}};\,\,18=2\int\limits_{2}^{6}{{f\left( x \right)dx;\,\,}}\int\limits_{2}^{6}{{f\left( x \right)dx=9\,\,}}\end{align}$

Definite Integral Trigonometry Problems

Problems: Evaluate the following (using what you know about Geometry and Trig Functions), given the fact that :

a. $ \displaystyle \int\limits_{0}^{{2\pi }}{{\sin \left( x \right)dx}}$      b. $ \displaystyle \int\limits_{0}^{{\frac{\pi }{2}}}{{\sin \left( x \right)dx}}$      c. $ \displaystyle \int\limits_{0}^{{2\pi }}{{\left| {\sin x} \right|\,dx}}$      d. $ \displaystyle \int\limits_{{-\frac{\pi }{2}}}^{{\frac{\pi }{2}}}{{\cos \left( x \right)dx}}$      e. $ \displaystyle \int\limits_{0}^{{\frac{\pi }{2}}}{{\left[ {\sin \left( x \right)+2x} \right]\,dx}}$

Solutions: 

a. Since $ \displaystyle \int\limits_{0}^{\pi }{{\sin \left( x \right)dx}}=2$, $ \displaystyle \int\limits_{0}^{{2\pi }}{{\sin \left( x \right)dx}}$ is this amount above the $ x$-axis (2), and also the opposite (negative) of this amount below the $ x$-axis (–2). Thus, we have $ 2+-2=0$.       

b. Since $ \displaystyle \int\limits_{0}^{\pi }{{\sin \left( x \right)dx}}=2$, $ \displaystyle \int\limits_{0}^{{\frac{\pi }{2}}}{{\sin \left( x \right)dx}}$ is half this amount, which is 1. (The function is symmetrical across $ \displaystyle x=\frac{\pi }{2}$).

c. Since $ \displaystyle \int\limits_{0}^{\pi }{{\sin \left( x \right)dx}}=2$, $ \displaystyle \int\limits_{0}^{{2\pi }}{{\left| {\sin \left( x \right)} \right|dx}}$ is this amount  above the $ x$-axis (2), and again above the $ x$-axis because of the absolute value. Thus, we have $ 2+2=4$.

 

d. (Horizontal Shift by $ \displaystyle \frac{\pi }{2}$ to Left, using Cosine) The graph of cosine is the graph of sin that is shifted either $ \displaystyle \frac{\pi }{2}$ to the left or right. Thus, $ \displaystyle \int\limits_{{-\frac{\pi }{2}}}^{{\frac{\pi }{2}}}{{\cos x\,dx}}=\int\limits_{0}^{\pi }{{\sin x\,dx}}$, which is $ 2$.

 

e. $ \displaystyle \int\limits_{0}^{{\frac{\pi }{2}}}{{\left[ {\sin \left( x \right)+2x} \right]\,dx}}=\int\limits_{0}^{{\frac{\pi }{2}}}{{\sin \left( x \right)dx}}+\int\limits_{0}^{{\frac{\pi }{2}}}{{2x\,dx}}$. We know that $ \displaystyle \int\limits_{0}^{{\frac{\pi }{2}}}{{\sin \left( x \right)dx}}=1$ from b. above. $ \displaystyle \int\limits_{0}^{{\frac{\pi }{2}}}{{2x\,dx}}$ is a triangle (see diagram) with a base of $ \displaystyle \frac{\pi }{2}$ and a height of $ \displaystyle 2\cdot \frac{\pi }{2}=\pi $, so its area is $ \displaystyle \frac{1}{2}bh=\frac{1}{2}\cdot \frac{\pi }{2}\cdot \pi =\frac{{{{\pi }^{2}}}}{4}$. The total area is $ \displaystyle 1+\frac{{{{\pi }^{2}}}}{4}\approx 3.47$.           

Mean Value Theorem (MVT) for Integrals

We learned about the Mean Value Theorem for Derivatives here in the Curve Sketching section. The Mean Value Theorem (MVT) for Integrals is a theorem that guarantees that a continuous function in an interval contains at least one point where that function is equal to the average value of the function.

What does this mean in plain English? All it really means is that for a continuous function between two different points, there is at least one point where the “$ y$” value is equal to the average of the all the “$ y$” values in that interval. In terms of geometry, this means that there exists a rectangle between the two points whose area is equal to the area under the curve of the function between those two points. Think of flattening a mountain so it fills the valleys perfectly.

Here’s the formal definition of the Mean Value Theorem for Integrals:

Mean Value Theorem for Integrals

For a function $ f$ that is continuous on closed interval $ [a,b]$, there exists at least one number $ c$ in that closed interval such that:

 

$ \displaystyle \,\,\,\,\,\,\,\,\,\,\,\int\limits_{a}^{b}{{f\left( x \right)dx=f\left( c \right)}}\left( {b-a} \right)$

 

For a geometric explanation, think of $ f\left( c \right)$ as the height of a rectangle and $ \left( {b-a} \right)$ as the width. There is a number $ c$ such that  has the same area as the region under the curve of that function from $ a$ to $ b$.

We’ll use this formula to solve problems where we find the “$ c$” guaranteed by the Mean Value Theorem for an integral in a specific interval. We’ll also derive the Average Value of a Function from this formula (below). Here are the types of problems you might see for the Mean Value Theorem:

Mean Value Theorem Problem Solution  
Find the value of “$ c$” that is guaranteed by the Mean Value Theorem (MVT) for integrals for $ f\left( x \right)=-{{x}^{2}}-8$ in the interval $ \left[ {1,4} \right]$. Use the MVT equation to find the “$ c$”:

$ \displaystyle \int\limits_{a}^{b}{{f\left( x \right)dx=f\left( c \right)}}\left( {b-a} \right):\,\,\,\,\,\,\int\limits_{1}^{4}{{\left( {-{{x}^{2}}-8} \right)dx=f\left( c \right)}}\left( {4-1} \right)$

$ \displaystyle \begin{align}\left[ {-\frac{{{{x}^{3}}}}{3}-8x} \right]_{1}^{4}&=\left( {-{{c}^{2}}-8} \right)\left( 3 \right)\\\left[ {-\frac{{{{4}^{3}}}}{3}-8\left( 4 \right)} \right]-\left[ {-{{{\frac{1}{3}}}^{3}}-8\left( 1 \right)} \right]&=3\left( {-{{c}^{2}}-8} \right)\end{align}$

$ -45=-3{{c}^{2}}-24;\,\,\,\,\,{{c}^{2}}=7;\,\,\,\,\,c=\pm \,\sqrt{7};\,\,\,c=\sqrt{7}$

Take positive only since the “$ c$” need to be in the interval $ \left[ {1,4} \right]$.

Find the value of “$ c$” that is guaranteed by the Mean Value Theorem (MVT) for integrals for $ f\left( x \right)=3{{\sec }^{2}}\left( x \right)$ in the interval $ \displaystyle \left[ {-\frac{\pi }{4},\frac{\pi }{4}} \right]$. Use the MVT equation to find the “$ c$”:

$ \displaystyle \int\limits_{a}^{b}{{f\left( x \right)dx=f\left( c \right)}}\left( {b-a} \right):\,\,\,\,\,\,\int\limits_{{-\frac{\pi }{4}}}^{{\frac{\pi }{4}}}{{\left( {3{{{\sec }}^{2}}x} \right)dx=f\left( c \right)}}\left[ {\frac{\pi }{4}-\left( {-\frac{\pi }{4}} \right)} \right]$

$ \displaystyle \left[ {3\tan x} \right]_{{-\frac{\pi }{4}}}^{{\frac{\pi }{4}}}=\left( {3{{{\sec }}^{2}}c} \right)\left[ {\frac{\pi }{4}-\left( {-\frac{\pi }{4}} \right)} \right]$

$ \displaystyle \left[ {3\tan \left( {\frac{\pi }{4}} \right)} \right]-\left[ {3\tan \left( {-\frac{\pi }{4}} \right)} \right]=\left( {3{{{\sec }}^{2}}c} \right)\left( {\frac{\pi }{2}} \right)$

$ \displaystyle \begin{align}3\cdot 1-\left( {3\cdot -1} \right)=\frac{{3\pi }}{2}{{\sec }^{2}}c;\,\,\,\,\,6=\frac{{3\pi }}{2}{{\sec }^{2}}c;\,\,\,\,\,\,\,{{\sec }^{2}}c=\frac{4}{\pi }\\{{\cos }^{2}}c=\frac{\pi }{4};\,\,\,\,\,\,\,\cos c=\pm \,\sqrt{{\frac{\pi }{4}}};\,\,\,\,\,\,\,\,c=\pm \,\arccos \sqrt{{\frac{\pi }{4}}};\,\,\,\,\,c=\pm \,.482\end{align}$

Use both positive and negative, since both values are in the interval $ \displaystyle \left[ {-\frac{\pi }{4},\frac{\pi }{4}} \right]$.

Find the value of “$ c$” that is guaranteed by the Mean Value Theorem (MVT) for integrals for $ f\left( x \right)=\ln \left( {x+2} \right)$ in the interval $ \left[ {-1,\,4} \right]$.

(Use calculator to get the integral of the ln function.)

Use the MVT equation to find the “$ c$”:

$ \displaystyle \int\limits_{a}^{b}{{f\left( x \right)dx=f\left( c \right)}}\left( {b-a} \right):\,\,\,\,\,\,\int\limits_{{-1}}^{4}{{\ln \left( {x+2} \right)dx=f\left( c \right)}}\left[ {4-\left( {-1} \right)} \right]$

$ \displaystyle 5.7506\approx \ln \left( {c+2} \right)\left( 5 \right);\,\,\,\,1.15012\approx \ln \left( {c+2} \right);\,\,\,\,\,\,{{e}^{{1.15012}}}\approx c+2;\,\,\,\,\,\,c\approx \,\,1.1586$

Average Value of a Function

Now we can solve for the Average Value of a Function by dividing both sides of the Mean Value Theorem equation by $ \left( {b-a} \right)$. The average value of a function is the $ f\left( c \right)$ in the Mean Value Theorem equation.

Average Value of a Function

If a function $ f$ is integrable on a closed interval $ [a,b]$, then the average value on that interval is:

$ \displaystyle \frac{1}{{\left( {b-a} \right)}}\int\limits_{a}^{b}{{f\left( x \right)dx}}$

To remember this, think Integral over Interval:

$ \displaystyle \text{Average Value}=\frac{{\int\limits_{a}^{b}{{f\left( x \right)dx}}}}{{\left( {b-a} \right)}}=\,\,\frac{{\text{Integral}}}{{\text{Interval}}}$

Here are some Average Value Theorem problems:

Average Value Theorem Problem Solution  
Find the Average Value of $ \displaystyle f\left( x \right)=2+\frac{8}{{{{x}^{2}}}}$ from $ \left[ {2,4} \right]$. Think “Integral Over Interval”:      Average Value = $ \displaystyle \frac{1}{{\left( {b-a} \right)}}\int\limits_{a}^{b}{{f\left( x \right)dx}}$

$ \displaystyle \begin{align}\frac{1}{{\left( {4-2} \right)}}\int\limits_{2}^{4}{{\left( {2+\frac{8}{{{{x}^{2}}}}} \right)dx}}&=\frac{1}{2}\int\limits_{2}^{4}{{\left( {2+8{{x}^{{-2}}}} \right)dx}}=\frac{1}{2}\left[ {2x-8{{x}^{{-1}}}} \right]_{2}^{4}\\&=\frac{1}{2}\left( {\left[ {2\left( 4 \right)-\frac{8}{4}} \right]-\left[ {2\left( 2 \right)-\frac{8}{2}} \right]} \right)=\frac{1}{2}\left( {8-2-4+4} \right)=3\end{align}$

Find the Average Value of $ f\left( x \right)=\cos x+3$ from $ \displaystyle \left[ {-\frac{\pi }{2},\,\frac{\pi }{2}} \right]$. Think “Integral Over Interval”:      Average Value = $ \displaystyle \frac{1}{{\left( {b-a} \right)}}\int\limits_{a}^{b}{{f\left( x \right)dx}}$

$ \displaystyle \begin{align}\frac{1}{{\left[ {\frac{\pi }{2}-\left( {-\frac{\pi }{2}} \right)} \right]}}\,\int\limits_{{-\frac{\pi }{2}}}^{{\frac{\pi }{2}}}{{\left( {\cos x+3} \right)dx}}&=\frac{1}{\pi }\int\limits_{{-\frac{\pi }{2}}}^{{\frac{\pi }{2}}}{{\left( {\cos x+3} \right)dx}}=\frac{1}{\pi }\left[ {\sin x+3x} \right]_{{-\frac{\pi }{2}}}^{{\,\,\,\frac{\pi }{2}}}\\&=\frac{1}{\pi }\left( {\left[ {\sin \left( {\frac{\pi }{2}} \right)+3\left( {\frac{\pi }{2}} \right)} \right]-\left[ {\sin \left( {-\frac{\pi }{2}} \right)+3\left( {-\frac{\pi }{2}} \right)} \right]} \right)\\&=\frac{1}{\pi }\left[ {1+\frac{{3\pi }}{2}-\left( {-1} \right)+\frac{{3\pi }}{2}} \right]=\frac{1}{\pi }\left( {2+3\pi } \right)=\,\,\frac{2}{\pi }+3\end{align}$

Find the number(s) $ b$ such that the Average Value of $ f\left( x \right)=6x-3$ on interval $ \left[ {0,b} \right]$ is 15. Think “Integral Over Interval”:      Average Value = $ \displaystyle \frac{1}{{\left( {b-a} \right)}}\int\limits_{a}^{b}{{f\left( x \right)dx}}$

$ \displaystyle \frac{1}{{b-0}}\int\limits_{0}^{b}{{\left( {6x-3} \right)dx}}=\frac{1}{b}\left[ {3{{x}^{2}}-3x} \right]_{0}^{b}$

$ \displaystyle \frac{1}{b}\left( {\left[ {3{{b}^{2}}-3b} \right]-\left[ {3{{{\left( 0 \right)}}^{2}}-3\left( 0 \right)} \right]} \right)=15;\,\,\,\,\,\frac{1}{b}\left( {3{{b}^{2}}-3b} \right)=15$

$ \displaystyle 3b-3=15;\,\,\,\,3b=18;\,\,\,\,\,b=6$

Here is an application of the Average Value:

Average Value Theorem Application Solution  
The temperature over a 10-hour period can be modeled by the function $ f\left( t \right)=-{{t}^{2}}+4t+45$.

 

a. Find the average temperature.

b. Find the minimum temperature.

c. Find the maximum temperature.

a. The average value over the 10-hour interval (using the calculator to find the integral) is:

$ \displaystyle \frac{1}{{\left( {b-a} \right)}}\int\limits_{a}^{b}{{f\left( t \right)dt}}=\frac{1}{{\left( {10-0} \right)}}\int\limits_{0}^{{10}}{{\left( {-{{t}^{2}}+4t+45} \right)dt}}=\left( {\frac{1}{{10}}} \right)316\frac{2}{3}=31\frac{2}{3}$ degrees.

To get the minimum and maximum temperatures over that period, graph, since we have to check the endpoints, too. (See here in the Introductions to Quadratics section for more information on how to find the minimum and maximum of quadratics using the graphing calculator.)

b. From the graph above, in the interval $ 0\le t\le 10$, the minimum temperature is –15 degrees, at the endpoint when $ t=10$.

c. From the graph above, in the interval $ 0\le t\le 10$, the maximum temperature is 49 degrees.

2nd Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus has to do with taking the derivative of the definite integral; we basically “undo” the integral to get the original function back. We have to be careful though, since we don’t always get the original function exactly as it was: it turns out (because of the Chain Rule), we have to multiply the function by the derivative of the upper limit of the interval. Also, to use this theorem, the lower bound must be a constant (although, if it’s not, it can be made into two integrals).

This theorem says that $ \displaystyle \frac{d}{{dx}}\left( {\int\limits_{a}^{x}{{f\left( t \right)dt}}} \right)=f\left( x \right)$, or if $ \displaystyle F\left( x \right)=\int\limits_{a}^{x}{{f\left( t \right)dt}}$, then $ {F}’\left( x \right)=f\left( x \right)$.

It looks like the derivative of an integral (accumulation function) gets us back to the original integrand with just a change of variables. But let’s keep going, using the First Fundamental Theorem of Calculus:  $ \displaystyle \frac{d}{{dx}}\left[ {\int\limits_{a}^{{g\left( x \right)}}{{f\left( t \right)\,dt}}} \right]=\frac{d}{{dx}}\left[ {F\left( {g\left( x \right)} \right)-F\left( a \right)} \right]=\frac{d}{{dx}}F\left( {g\left( x \right)} \right)=f\left( {g\left( x \right)} \right)\cdot {g}’\left( x \right)$, where $ a$ is a constant. The $ F\left( a \right)$ disappears, since the derivative of a constant is 0, but for the $ F\left( x \right)$, we substitute the upper limit in for the variable, but then have to use the chain rule to multiply by the derivative of this function.

Here’s an example, where we solve the definite integral and then take the derivative back. Note that you won’t have to do this much work for each problem; we’ll see soon that we just substitute the upper limit in the integral function, and then multiply by the derivative of that upper limit.

Find the derivative of:    $ \displaystyle F\left( x \right)=\int\limits_{\pi }^{{2{{x}^{3}}}}{{\sin \left( t \right)dt}}$:

First solve for the integral:  $ \displaystyle \int\limits_{\pi }^{{2{{x}^{3}}}}{{\sin \left( t \right)\,dt=\left[ {-\cos \left( t \right)} \right]}}_{\pi }^{{2{{x}^{3}}}}=-\cos \left( {2{{x}^{3}}} \right)-\left[ {-\cos \left( \pi \right)} \right]=-\cos \left( {2{{x}^{3}}} \right)-1$

Then take the derivative:  $ \displaystyle \frac{d}{{dx}}\left[ {-\cos \left( {2{{x}^{3}}} \right)-1} \right]=\frac{d}{{dx}}\left[ {-\cos \left( {2{{x}^{3}}} \right)} \right]-\frac{d}{{dx}}\left( 1 \right)=\sin \left( {2{{x}^{3}}} \right)\cdot 6{{x}^{2}}-0=6{{x}^{2}}\sin \left( {2{{x}^{3}}} \right)$

Here’s the formal definition of the 2nd Fundamental Theorem of Calculus, with the basic instructions on how to solve these problems:

2nd Fundamental Theorem of Calculus

If $ f$ is continuous on an open interval that contains $ a$ (a constant), for every $ x$ in the interval,

If $ \displaystyle F\left( x \right)=\int\limits_{a}^{x}{{f\left( t \right)dt}}$, then $ {F}’\left( x \right)=f\left( x \right)$            (alternate way: $ \displaystyle \frac{d}{{dx}}\left[ {\int\limits_{a}^{x}{{f\left( t \right)}}F\left( t \right)} \right]=f\left( x \right)$)

When finding $ {F}’\left( x \right)$, plug in the upper bound into the function $ f\left( t \right)$ directly, but if the upper bound is different than just plain “$ x$”, multiply the function by the derivative of the upper bound.

In other words, if $ \displaystyle F\left( x \right)=\int\limits_{a}^{{g\left( x \right)}}{{f\left( t \right)dt}}$, then $ {F}’\left( x \right)=f\left( {g\left( x \right)} \right)\cdot {g}’\left( x \right)$. (For example, if the upper bound is $ {{x}^{2}}$, plug in the $ {{x}^{2}}$ everywhere there’s a $ t$, but then multiply by $ 2x$, the derivative of $ {{x}^{2}}$).

If there are variables in both the upper and lower bounds, separate the integral into two integrals with constants. For example, separate $ \displaystyle \int\limits_{{f\left( x \right)}}^{{g\left( x \right)}}{{}}$ into $ \displaystyle \int\limits_{{f\left( x \right)}}^{a}{{}}$ (which is  $ \displaystyle -\int\limits_{a}^{{f\left( x \right)}}{{}}$) and $ \displaystyle \int\limits_{a}^{{g\left( x \right)}}{{}}$.

Here are some Second Theorem of Calculus problems:

2nd Theorem of Calculus Problem Solution  
Find the derivative of

$ \displaystyle F\left( x \right)=\int\limits_{{-3}}^{x}{{\left( {{{t}^{3}}-4t} \right)dt}}$

$ \displaystyle {F}’\left( x \right)=\left( {{{x}^{3}}-4x} \right)\cdot \frac{d}{{dx}}\left( x \right)=\left( {{{x}^{3}}-4x} \right)\cdot 1={{x}^{3}}-4x$

Note that since the upper limit is just $ x$, the derivative is the same as the original integrand.

Find the derivative of

$ \displaystyle F\left( x \right)=\int\limits_{2}^{{{{x}^{4}}}}{{\frac{1}{{{{t}^{2}}}}\,dt}}$.

$ \displaystyle {F}’\left( x \right)=\frac{1}{{{{{\left( {{{x}^{4}}} \right)}}^{2}}}}\cdot \frac{d}{{dx}}\left( {{{x}^{4}}} \right)=\frac{1}{{{{x}^{8}}}}\left( {4{{x}^{3}}} \right)=4{{x}^{{-5}}}=\frac{4}{{{{x}^{5}}}}$

Note:  Let’s show how this works “the long way”:

$ \displaystyle \int\limits_{2}^{{{{x}^{4}}}}{{\frac{1}{{{{t}^{2}}}}\,dt}}=\int\limits_{2}^{{{{x}^{4}}}}{{{{t}^{{-2}}}\,dt}}=\left[ {-{{t}^{{-1}}}} \right]_{2}^{{{{x}^{4}}}}=-{{\left( {{{x}^{4}}} \right)}^{{-1}}}-\left( {-{{2}^{{-1}}}} \right)=-\frac{1}{{{{x}^{4}}}}+\frac{1}{2}$

Now take the derivative back:

$ \displaystyle \frac{d}{{dx}}\left( {-\frac{1}{{{{x}^{4}}}}+\frac{1}{2}} \right)=\frac{d}{{dx}}\left( {-{{x}^{{-4}}}} \right)+\frac{d}{{dx}}\left( {\frac{1}{2}} \right)=4{{x}^{{-5}}}+0=\frac{4}{{{{x}^{5}}}}$   √

Find the derivative of

$ \displaystyle F\left( x \right)=\int\limits_{{-\pi }}^{{{{x}^{2}}}}{{\sin {{t}^{2}}\,dt}}$

$ \displaystyle {F}’\left( x \right)=\sin \left[ {{{{\left( {{{x}^{2}}} \right)}}^{2}}} \right]\cdot \frac{d}{{dx}}\left( {{{x}^{2}}} \right)=\sin \left( {{{x}^{4}}} \right)\cdot 2x=2x\sin {{x}^{4}}$
 

Find the derivative of

$ \displaystyle F\left( x \right)=\int\limits_{x}^{{3x}}{{2{{t}^{3}}\,dt}}$

Since we don’t have a constant for the lower bound, we need to separate the integral into two integrals, each with a constant. Remember that we can switch the lower and upper bounds in an integral by taking the opposite (the negative) of that integral:

$ \displaystyle \int\limits_{x}^{{3x}}{{2{{t}^{3}}\,dt}}=\int\limits_{x}^{a}{{2{{t}^{3}}\,dt}}+\int\limits_{a}^{{3x}}{{2{{t}^{3}}\,dt}}=-\int\limits_{a}^{x}{{2{{t}^{3}}\,dt}}+\int\limits_{a}^{{3x}}{{2{{t}^{3}}\,dt}}$

$ \begin{align}{F}’\left( x \right)&=-2{{x}^{3}}\cdot \frac{d}{{dx}}\left( x \right)+2{{\left( {3x} \right)}^{3}}\cdot \frac{d}{{dx}}\left( {3x} \right)\\&=-2{{x}^{3}}\cdot 1+2\left( {27{{x}^{3}}} \right)\cdot 3=-2{{x}^{3}}+162{{x}^{3}}=160{{x}^{3}}\end{align}$

Using U-Substitution with Definite Integration

We learned how to use U-substitution (U-Sub) here in the U-Substitution Integration section, but let’s do some problems using Definite Integration. Remember this about U-sub Indefinite Integration:

U-Substitution Integration

 

Outside Function            Derivative of Inside Function

$ \displaystyle \int{{\color{red}{{f\left( {\color{green}{{g\left( x \right)}}} \right)}}\,\color{purple}{{{g}’\left( x \right)}}}}\,dx=\color{blue}{{F\left( {g\left( x \right)} \right)}}\,+\,C$

Inside Function              Any Composite Function

What this says is that if we want the integral of the outside function, to make it work, we have to make sure that what we’re integrating somehow has a factor that is the derivative of the inside function. (We can “trick” the integrand into having this factor.)

With U-sub and Definite Integration, we can do these problems in one of two ways:

  • Substitute the expression for “$ u$” back and use the original values for the upper and lower bounds. Remember that the upper and lower bounds are in terms of “$ x$”.
  • Keep the “$ u$” in the expression and solve for new upper and lower bounds (solve for “$ x$” in terms of “$ u$”). Then we don’t have to put the expression for “$ u$” back in the problem! (We only want to do this if it’s straightforward to get “$ x$” in terms of “$ u$”.)

We’ll show both these methods; the main thing is to make sure your upper and lower bounds match the variable you’re plugging them in for:

Definite Integral U-Substitution Problem Solution  
Find the integral:

$ \displaystyle \int\limits_{{-2}}^{2}{{x{{{\left( {{{x}^{2}}-3} \right)}}^{2}}\,dx}}$

$ \displaystyle \begin{align}&\color{blue}{{u={{x}^{2}}-3}}\\&du=2x\,dx\\&\color{green}{{dx=\frac{{du}}{{2x}}}}\end{align}$           $ \require {cancel} \displaystyle \int_{{-2}}^{2}{{x{{{\left( {\color{blue}{{{{x}^{2}}-3}}} \right)}}^{2}}}}\color{green}{{dx}}=\int_{{-2}}^{2}{{\cancel{x}\cdot {{{\color{blue}{u}}}^{2}}}}\color{green}{{\frac{{du}}{{2\cancel{x}}}}}=\int_{{-2}}^{2}{{\frac{{{{{\color{blue}{u}}}^{2}}}}{2}\color{green}{{du}}}}=\left[ {\frac{{{{u}^{3}}}}{6}} \right]_{{x=-2}}^{{x=2}}$

Substitute back in for “$ u$” since it’s not easy to solve for $ x$ in terms of $ u$:

$ \displaystyle \left[ {\frac{{{{u}^{3}}}}{6}} \right]_{{x=-2}}^{{x=2}}=\left[ {\frac{{{{{\left( {{{x}^{2}}-3} \right)}}^{3}}}}{6}} \right]_{{-2}}^{2}=\frac{{{{{\left( {{{2}^{2}}-3} \right)}}^{3}}}}{6}-\frac{{{{{\left[ {{{{\left( {-2} \right)}}^{2}}-3} \right]}}^{3}}}}{6}=\frac{1}{6}-\frac{1}{6}=0$

Find the integral:

$ \displaystyle \int\limits_{{-\frac{\pi }{2}}}^{{\frac{\pi }{2}}}{{{{{\sin }}^{3}}\left( x \right)\cos \left( x \right)dx}}$

$ \displaystyle \begin{array}{l}\\\color{blue}{{u=\sin \left( x \right)}}\\du=\cos \left( x \right)dx\\\color{green}{{dx=\frac{{du}}{{\cos \left( x \right)}}}}\end{array}$          $ \displaystyle \begin{align}&\int\limits_{{-\frac{\pi }{2}}}^{{\frac{\pi }{2}}}{{{{{\color{blue}{{\sin }}}}^{3}}\color{blue}{{\left( x \right)}}\cos \left( x \right)\,\color{green}{{dx}}}}=\int\limits_{{-\frac{\pi }{2}}}^{{\frac{\pi }{2}}}{{{{{\color{blue}{u}}}^{3}}\,\cancel{{\cos \left( x \right)}}\,\color{green}{{\frac{{du}}{{\cancel{{\cos \left( x \right)}}}}}}}}=\int\limits_{{-\frac{\pi }{2}}}^{{\frac{\pi }{2}}}{{{{{\color{blue}{u}}}^{3}}\color{green}{{du}}}}\\&=\left[ {\frac{{{{u}^{4}}}}{4}} \right]_{{x=-\frac{\pi }{2}}}^{{x=\frac{\pi }{2}}}=\left[ {\frac{{{{{\sin }}^{4}}\left( x \right)}}{4}} \right]_{{-\frac{\pi }{2}}}^{{\frac{\pi }{2}}}=\frac{{{{{\sin }}^{4}}\left( {\frac{\pi }{2}} \right)}}{4}-\frac{{{{{\sin }}^{4}}\left( {-\frac{\pi }{2}} \right)}}{4}=0\end{align}$
Find the integral:

$ \displaystyle \int\limits_{1}^{4}{{\frac{2}{{\sqrt{x}\,{{{\left( {1+\sqrt{x}} \right)}}^{2}}}}}}$

$ \displaystyle \begin{align}&\color{blue}{{u=1+\sqrt{x}=1+{{x}^{{\frac{1}{2}}}}}}\\&du=\frac{1}{2}{{x}^{{-\frac{1}{2}}}}dx\\&\color{green}{{dx=\,\frac{{du}}{{\frac{1}{2}{{x}^{{-\frac{1}{2}}}}}}=2{{x}^{{\frac{1}{2}}}}\,du}}\end{align}$              $ \displaystyle \begin{align}&\int\limits_{1}^{4}{{\frac{2}{{\sqrt{x}\,{{{\left( {\color{blue}{{1+\sqrt{x}}}} \right)}}^{2}}}}}}\,\color{green}{{dx}}=\int\limits_{1}^{4}{{\frac{2}{{\sqrt{x}\,\color{blue}{{{{u}^{2}}}}}}}}\,\color{green}{{dx}}=\int\limits_{1}^{4}{{\frac{2}{{\cancel{{{{x}^{{\frac{1}{2}}}}}}\,\color{blue}{{{{u}^{2}}}}}}}}\color{green}{{\,\left( {2\cancel{{{{x}^{{\frac{1}{2}}}}}}\,du} \right)}}=\int\limits_{1}^{4}{{4\color{blue}{{{{u}^{{-2}}}}}\color{green}{{du}}}}\\&=\left[ {\frac{{4{{u}^{{-1}}}}}{{-1}}} \right]_{{x=1}}^{{x=4}}=\left[ {\frac{{-4}}{u}} \right]_{{x=1}}^{{x=4}}=\left[ {\frac{{-4}}{{1+\sqrt{x}}}} \right]_{1}^{4}=\left( {\frac{{-4}}{{1+\sqrt{4}}}} \right)-\left( {\frac{{-4}}{{1+\sqrt{1}}}} \right)=\frac{2}{3}\end{align}$
Find the integral:

$ \displaystyle \int\limits_{{-1}}^{2}{{x\sqrt{{x+2}}\,dx}}$

For this problem, find the integral by replacing the upper and lower bounds with value of “$ u$”.

$ \displaystyle \begin{array}{l}\color{blue}{{u=x+2}}\\\color{red}{{x=u-2}}\\du=dx\\\color{green}{{dx=du}}\end{array}$              $ \displaystyle \begin{align}&\int_{{-1}}^{2}{{\color{red}{x}\sqrt{{\color{blue}{{x+2}}}}\,\color{green}{{dx}}=\int_{{-1}}^{2}{{\color{red}{x}{{{\left( {\color{blue}{{x+2}}} \right)}}^{{\frac{1}{2}}}}\,\color{green}{{dx}}=}}}}\int_{{-1}}^{2}{{\color{red}{{\left( {u-2} \right)}}\cdot {{{\color{blue}{u}}}^{{\frac{1}{2}}}}}}\color{green}{{du}}=\int_{{-1}}^{2}{{\left( {\color{blue}{{u\cdot {{u}^{{\frac{1}{2}}}}-2{{u}^{{\frac{1}{2}}}}}}} \right)}}\,\color{green}{{du}}\\&=\int_{{-1}}^{2}{{\left( {\color{blue}{{{{u}^{{\frac{3}{2}}}}-2{{u}^{{\frac{1}{2}}}}}}} \right)}}\,\color{green}{{du}}=\left[ {\frac{{{{u}^{{\frac{5}{2}}}}}}{{\frac{5}{2}}}-2\frac{{{{u}^{{\frac{3}{2}}}}}}{{\frac{3}{2}}}} \right]_{{u=x+2=-1+2=1}}^{{u=x+2=2+2=4}}=\left[ {\frac{{2{{u}^{{\frac{5}{2}}}}}}{5}-\frac{{4{{u}^{{\frac{3}{2}}}}}}{3}} \right]_{1}^{4}\\\,\,\,\,\,\,\,&=\left[ {\frac{{2{{{\left( 4 \right)}}^{{\frac{5}{2}}}}}}{5}-\frac{{4{{{\left( 4 \right)}}^{{\frac{3}{2}}}}}}{3}} \right]-\left[ {\frac{{2{{{\left( 1 \right)}}^{{\frac{5}{2}}}}}}{5}-\frac{{4{{{\left( 1 \right)}}^{{\frac{3}{2}}}}}}{3}} \right]=\frac{{46}}{{15}}\end{align}$

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On to Integration as Accumulated Change– you’re ready!