Volume of a Solid of Revolution in Polar Coordinates Example
\[V= \pi \int^{x_2}_{x_1} y^2dx= \pi \int^{\theta_2}_{\theta_1} (y( \theta ))^2 \frac{dx}{d \theta} d \theta\]
.Example: Find the volume generated by rotating the hypocycloid
\[x=a cos^3 \theta , \; y= a sin^3 \theta\]
in the first quadrant about the \[x\]
axis..The limits of
\[\theta\]
are 0 and \[ \frac{ \pi}{2}\]
. Then\[\begin{equation} \begin{aligned} V &= \pi \int^0_{\frac{\pi}{2}} (a sin^3 \theta )^2 (-3acos^2 \theta sin \theta ) d \theta \\ &= -3a^3\pi \int^0_{\frac{\pi}{2}} sin^7 \theta cos^2 \theta d \theta \\ &= -3a^3 \pi \int^0_{\frac{\pi}{2}} sin^7 \theta (1-sin^2 \theta ) d \theta \\ &= -3a^3 \int^0_{\frac{\pi}{2}} sin^7 \theta -sin^9 \theta d \theta \\ &= 3a^3 \pi (\frac{6 \times 4 \times 2}{7 \times 5 \times 3 \times 1}- \frac{8 \times 6 \times 4 \times 2}{9 \times 7 \times 5 \times 3 \times 1} ) \\ &= \frac{16 \pi }{105} \end{aligned} \end{equation}\]
