How to find the area inside the larger loop and outside the smaller loopof the limacon $r=\frac{1}{2} +\cos \theta$?
Once the integrals are set up, I know how to solve them, but I'm having difficulty setting the integrals up.
calculusintegrationpolar coordinates
How to find the area inside the larger loop and outside the smaller loopof the limacon $r=\frac{1}{2} +\cos \theta$?
Once the integrals are set up, I know how to solve them, but I'm having difficulty setting the integrals up.
Best Answer
The area of one half of the outer loop is given by $\displaystyle\frac{1}{2} \int \limits_{0}^{2\pi/3} (\frac{1}{2} +\cos \theta)^2 d\theta$. See This.
The area of one half of the inside loop is given by $\displaystyle\frac{1}{2} \int \limits_{\pi}^{4\pi/3} (\frac{1}{2} +\cos \theta)^2 d\theta$. See This.
Now, we subtract the outer loop from the inner loop and multiply by $2$ to account for symmetry:
$$2\left(\displaystyle\frac{1}{2} \int \limits_{0}^{2\pi/3} (\frac{1}{2} +\cos \theta)^2 d\theta-\displaystyle\frac{1}{2} \int \limits_{\pi}^{4\pi/3} (\frac{1}{2} +\cos \theta)^2 d\theta\right)$$