# Math Help - Areas in polar coordinates

1. ## Areas in polar coordinates

I don't mean to post more than one math problem in a thread but I've been trying to work on these problems and other sites haven't been giving adequate feedback..., if anyone can also explain how you choose your limits of integration?

Find the area of it encloses.
1.) r = 5(1 + cos(θ))

2.) Find the area of the region enclosed by one loop of the curve.
r = sin(10θ)

3.) Find the area of the region that lies inside both curves.
r = 4 + 3sin(θ)
r = 4 + 3cos(θ)

2. Originally Posted by melmel
I don't mean to post more than one math problem in a thread but I've been trying to work on these problems and other sites haven't been giving adequate feedback..., if anyone can also explain how you choose your limits of integration?

Find the area of it encloses.
1.) r = 5(1 + cos(θ))
You could try sketching the curve, or just look at what happens as $\theta$ varies over a period of $\cos$. Anyway you will find that the natural limits of integration for this are $\theta=-\pi$ and $\theta=\pi$. Then the area:

$A=\int_{\theta=-\pi}^{\pi} r ~d\theta=\int_{\theta=-\pi}^{\pi} 5(1-\cos(\theta) ~d\theta$

RonL

3. Originally Posted by melmel
I don't mean to post more than one math problem in a thread but I've been trying to work on these problems and other sites haven't been giving adequate feedback..., if anyone can also explain how you choose your limits of integration?

2.) Find the area of the region enclosed by one loop of the curve.
r = sin(10θ)
One loop of the curve is generated when $10\theta$ varies over a range for which $\sin(10 \theta) \ge 0$ and is zero at both end points. Such a range is $(0, \pi/10).$

RonL