| Introduction and Net Change Theorem | Integration as Accumulated Change Problems |
| Integration as Accumulated Change Hints |
Integration as Accumulated Change
As the result of the Fundamental Theorems of Calculus from the Definite Integration section, we can now use integration to solve Accumulated Rate of Change, or Net Change problems. Here is the Net Change Theorem (which is basically a reinstatement of the Fundamental Theorem of Calculus):
The Accumulated Rate of Change can be measured by the area under the graph of a function over a certain interval. Thus, this can be represented by the definite integral of the function. We must be careful though, since the area below the $ \boldsymbol {x}$-axis is considered to be negative in measuring accumulated change.
(Note that we also addressed Position, Velocity, and Acceleration with Derivatives here in the Equation of the Tangent Line, Tangent Line Approximation, and Rates of Change section, and here in the Antiderivatives and Indefinite Integration section.)
Important Hint for Definite Integration Applications: If you’re not sure about whether to integrate, or what to integrate, remember this: the area under the curve is the integral. That area will represent different things, based on what the units of the axes are. In general, you just need to multiply the units of the $ y$-axis and the units of the $ x$-axis to get the units of the area under the curve. For example, If the $ y$-axis represents velocity and the $ x$-axis represents time, the integral represents total distance. As another example, if the $ y$-axis represents calories/day and the $ x$-axis represents days, then the area under the curve would represent the total calories over the days specified.
Integration as Accumulated Change Hints
When working these problems, remember the following:
Integration as Accumulated Change Problems
First, here’s a problem that uses the area under a curve:
Here’s another type of problem you may see:
Here are more Integration as Accumulated Change problems. Note that in some problems, we will use the fnInt( function on our calculator to integrate. (We will learn how to integrate exponents here in the Exponential and Logarithmic Integration section.)
On to Exponential and Logarithmic Integration – you’re ready!