Question

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

Answer 1

The region bounded by the polar curves looks like:

Since the region consists of two identical leaves that are symmetric about the `y`-axis, I will try to find a half of one leaf then multiply it by `4`.

`A=4int_0^{pi/2}int_0^{1-cos theta}rdrd theta`

`=4int_0^{pi/2}[r^2/2]_0^{1-cos theta}d theta`

`=2int_0^{pi/2}(1-2cos theta+cos^2theta)d theta`

by `cos^2theta=1/2(1+cos2theta)`,

`=int_0^{pi/2}(3-4cos theta+cos2theta)d theta`

`=[3theta-4sintheta+1/2sin2theta]_0^{pi/2}`

`={3pi}/2-4`

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I hope that this was helpful.