AP Calculus BC / Unit 8: Applications of Integration
8.4 Finding the Area Between Curves Expressed as Functions of x
Notes
$$A=\int_a^b [\text{top}(x)-\text{bottom}(x)]\,dx$$
- Find intersection points to get the limits.
Practice
Area between $y=x$ and $y=x^2$ from 0 to 1:
$\displaystyle \int_0^1(x-x^2)dx=\frac{1}{2}-\frac{1}{3}$.
On $[0,1]$, which is the top curve for $\displaystyle y=\sqrt x$ and $y=x$?
For $0<x<1$, $\displaystyle \sqrt x>x$.
Area between $y=4$ and $y=x^2$:
$\displaystyle \int_{-2}^2(4-x^2)dx=\frac{32}{3}$.
Ready for more? Take the Unit 8 test.