AP Calculus BC / Unit 8: Applications of Integration
8.7 Volumes with Cross Sections: Squares and Rectangles
Notes
$$V=\int_a^b A(x)\,dx$$
- Square cross sections with side $s(x)$: $A=s^2$.
- Rectangles with base $s$ and height $h$: $A=s\cdot h$.
Practice
A base region lies between $y=x$ and the $x$-axis on $[0,2]$. Square cross sections perpendicular to the $x$-axis. Volume:
$\displaystyle \int_0^2x^2dx$.
For squares, the side length $s(x)$ is usually:
The slice spans the base region.
Rectangles with height twice the base $s$ have area:
$s\cdot2s$.
Ready for more? Take the Unit 8 test.