Question

How do you find the area between the loop of `r=1+2costheta`?

Answer 1

Please see the explanation.

Explanation:

This reference Area with Polar Coordinates does a very similar exercise.

The equation from the reference is:

`Area = int_alpha^beta 1/2r^2 d theta`

We know `r(theta)` but we need to find the value of `alpha and beta`

The sample problem tells us that the loop starts at: `theta = (2pi)/3` and it ends at `theta = (4pi)/3`

The integral for the area of the loop is

`Area = int_((2pi)/3)^((4pi)/3) 1/2(1 + 2cos(theta))^2d theta`

`Area = int_((2pi)/3)^((4pi)/3) 1/2(1 + 4cos(theta) + 4cos^2(theta))d theta`

`Area = int_((2pi)/3)^((4pi)/3) 1/2(3 + 4cos(theta) + 2cos(2theta))d theta`

`Area = 1/2(3theta + 4sin(theta) + sin(2theta))|_((2pi)/3)^((4pi)/3)`

I am going to let you do the evaluation.