# Thread: areas and lengths in polar coordinates

1. ## areas and lengths in polar coordinates

a few questions

1.) find the area enclosed by one loop of the curve r= cos 4 theta

2.) find the area enclosed by one loop of the curve r= 2sin(5theta)

3.) find the area of the region that lies inside the first curve and outside the second curve

r= 5cos(theta)
r= 2+cos(theta)

4.) find the area inside the larger loop and outside the smaller loop of the limacon r= root 3/2 (only the 3 in root) + cos(theta)

2. Hello, Gauss's law!

These are basically of the same type: . $A \;=\;\tfrac{1}{2}\int^{\beta}_{\alpha} r^2\,d\theta$

1) Find the area enclosed by one loop of the curve: . $r\:=\:\cos4\theta$
This is an 8-leaf rose curve with one leaf on the 0°-line.

To find the limits, solve $r = 0.$
. . $\cos4\theta \:=\:0 \quad\Rightarrow\quad 4\theta \:=\:\pm\tfrac{\pi}{2} \quad\Rightarrow\quad \theta \:=\:\pm\tfrac{\pi}{8}$

Due to its symmetry, we can integrate from 0 to $\tfrac{\pi}{8}$ and multiply by 2.

We have: . $A \;=\;2\times\tfrac{1}{2}\!\int^{\frac{\pi}{8}}_0 \cos^2\!4\theta\,d\theta \;=\;\tfrac{1}{2}\!\int^{\frac{\pi}{8}}_0 (1 + \cos8\theta)\,d\theta$

Can you finish it?

3. The step you stopped on is the step I have problems with actually.