Question

How do you find the area of the region bounded by the polar curve `r=3(1+cos(theta))` ?

Answer 1

The area of the region enclosed by the curve `r=3(1=cos theta)` can be found by the double integral
`int_0^{2pi}int_0^{3(1+cos theta)}rdrd theta=27/2pi`

Let us evaluate the double integral.
`int_0^{2pi}int_0^{3(1+cos theta)}rdrd theta`
by Power Rule,
`=int_0^{2pi}[r^2/2]_0^{3(1+cos theta)}d theta`
`=int_0^{2pi}{[3(1+cos theta)]^2}/2d theta`
`=9/2int_0^{2pi}(1+2cos theta+cos^2 theta)d theta`
by `cos^2 theta=1/2(1+cos 2theta)`,
`=9/2int_0^{2pi}(3/2+2cos theta+1/2cos2theta)d theta`
`=9/2[3/2theta+2sin theta+1/4sin2theta]_0^{2pi}`
`=9/2cdot3/2(2pi)=27/2pi`