AP Calculus BC / Unit 8: Applications of Integration
8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
Notes
$$V=\pi\int_a^b [R(x)]^2\,dx$$
- $R$ is the distance from the axis to the curve.
- For the $y$-axis, write $R$ in terms of $y$ and integrate $dy$.
Practice
Revolve $\displaystyle y=\sqrt x$ on $[0,4]$ around the $x$-axis. Volume:
$\displaystyle \pi\int_0^4x\,dx=8\pi$.
Revolve $y=x$ on $[0,3]$ around the $x$-axis (a cone). Volume:
$\displaystyle \pi\int_0^3x^2dx=9\pi$.
In the disc method, $R(x)$ is:
The radius of each disc.
Ready for more? Take the Unit 8 test.