Approximating Areas
Net Change Theorem - FTC6
Objectives
- Explain the significance of the Net Change Theorem
- Use the Net Change Theorem to solve applied problems
Summary
The net change theorem considers the integral of a rate of change. The net change theorem states that the integral of a rate of change will describe the net change between the given limits.
In essence, if we had a function describing the rate of change of a container filling up, \(C'(x)\) , with limits \(a\) and \(b\) describing time, then the amount of liquid put into the container between time \(a\) and time \(b\) can be found by doing the following $$\int_a^b C'(x) \,dx = C(b) - C(a)$$
In the setting where we consider the integral of a velocity function \(v\), measures the exact change in position of the moving object on when [\(a\),\(b\)]; when \(v\) is nonnegative, is the object's distance traveled on [\(a\),\(b\)].