Areas in polar coordinates

Unit 8.x of Math 152

How to sketch a polar graph

• ex.

  • $r=4\sin (3\theta )$
    • make a table
      | $\theta$ | R |
      | - | --- |
      | 0 | 0 |
      | $\frac{\pi }{6}$ | 4 |
      | $\frac{\pi }{3}$ | 0 |
      | $\frac{\pi }{2}$ | -4 |-

• Area

  • $A={\int }_0^{\pi }\frac{r^2}{2}d\theta$
    • => $u=3\theta ,du=-3d\theta$
      • => $\frac{1}{2}{\int }_0^{\pi }16{\sin }^{2}3\theta d\theta$
        • $\frac{8}{3}{\int }_0^{3\pi }{\sin }^{2}udu$

          • (using any sin reduction formula)

            • => $4\pi$