AP Calculus BC / Unit 8: Applications of Integration
8.8 Volumes with Cross Sections: Triangles and Semicircles
Notes
- Semicircle with diameter $s$: $\displaystyle A=\frac{\pi}{8}s^2$.
- Equilateral triangle with side $s$: $\displaystyle A=\frac{\sqrt3}{4}s^2$.
- Isosceles right triangle with leg $s$: $\displaystyle A=\frac{1}{2}s^2$.
Practice
A semicircle cross section has diameter 4. Its area is:
$\displaystyle \frac{\pi}{8}(16)$.
An equilateral triangle with side 2 has area:
$\displaystyle \frac{\sqrt3}{4}(4)$.
Semicircle cross sections with diameter $s=x$ on $[0,2]$ give volume:
$\displaystyle \frac{\pi}{8}\int_0^2x^2dx=\frac{\pi}{8}\cdot\frac{8}{3}$.
Ready for more? Take the Unit 8 test.