## itzmashy Group Title a. find the area enclosed in the inner loop of the limacon, r=1+2costheta. b. find hte area inside the limacon but outside the inner loop 2 years ago 2 years ago

1. eliassaab Group Title

b) $2 \int_0^{\frac{2 \pi }{3}} \frac{1}{2} (2 \cos (\theta )+1)^2 \, d\theta=\frac{3 \sqrt{3}}{2}+2 \pi$

2. eliassaab Group Title

a)$2 \int_{\frac{2 \pi }{3}}^{\pi } \frac{1}{2} (2 \cos (\theta )+1)^2 \, d\theta=\pi -\frac{3 \sqrt{3}}{2}$

3. itzmashy Group Title

i'm not supposed to use a double integral?

4. colorful Group Title

the way I see it the first step is to find the bounds on theta I think having the picture ion front of you helps

5. eliassaab Group Title

Both of them are single integrals

6. colorful Group Title
7. colorful Group Title

sorry those bounds are wrong $r=0=1+2\cos\theta\implies \theta=\frac{2\pi}3,\frac{4\pi}3$which means the bounds on theta are$\frac{2\pi}3\le\theta\le\frac{4\pi}3$

8. colorful Group Title

The bounds on r are just the polar function itself$0\le r\le1+2\cos\theta$and the area differential in polar coordinates is$dA=rdrd\theta$hence the intergal for the area of the inner loop should be$\large \int\int dA=\int_{\frac{2\pi}3}^{\frac{4\pi}3}\int_{0}^{1+2r\cos\theta}rdr\theta$

9. colorful Group Title

...I assumed we are working in the interval $$[0,2\pi]$$

10. itzmashy Group Title

and @colorful i'm at work so i can't be checking this all the time :( but it was a homework problem last night and it blew my mind cuz i've never even seen a limacon before last night.. -_-"

11. eliassaab Group Title

The bounds for my solutions above are fine