AP Calculus AB / Unit 8: Applications of Integration
8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
Notes
$$V=\pi\int_a^b\left([R(x)]^2-[r(x)]^2\right)dx$$
- $R$: outer radius (far curve). $r$: inner radius (near curve).
Practice
Region between $y=x$ and $y=x^2$ revolved around the $x$-axis. Outer radius:
On $[0,1]$, $y=x$ is farther from the axis.
The volume for that region is:
$\displaystyle \pi\int_0^1(x^2-x^4)dx=\pi\left(\frac{1}{3}-\frac{1}{5}\right)$.
A washer is used instead of a disc when:
A gap creates a hole.
Ready for more? Take the Unit 8 test.