# How do I find the surface area of the solid defined by revolving r = 3sin(theta) about the polar axis?

Feb 24, 2015

Write :

$r = 3 \sin \left(\theta\right)$

$r = 3 \frac{y}{r}$ because $y = r \sin \left(t\right)$

${r}^{2} = 3 y$

${x}^{2} + {y}^{2} = 3 y$ because $r = \sqrt{{x}^{2} + {y}^{2}}$.

${x}^{2} + {\left(y - \frac{3}{2}\right)}^{2} = \frac{9}{4}$

You recognize a circle of radius $\frac{3}{2}$. The area is $\pi {\left(\frac{3}{2}\right)}^{2} = 9 \frac{\pi}{4}$.