Question

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

Answer 1

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

Answer 2

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

\]

Answer 3

i'm not supposed to use a double integral?

Answer 4

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

Answer 5

Both of them are single integrals

Answer 6

http://www.wolframalpha.com/input/?i=plot%20r%3D1%2B2costheta.%20%20&t=crmtb01

Answer 7

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\[\frac{2\pi}3\le\theta\le\frac{4\pi}3\]

Answer 8

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\[\large \int\int dA=\int_{\frac{2\pi}3}^{\frac{4\pi}3}\int_{0}^{1+2r\cos\theta}rdr\theta\]

Answer 9

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

Answer 10

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.. -_-"

Answer 11

The bounds for my solutions above are fine