A Conceptual Polar Curve Question

Question

A polar curve has $r=f(\theta), 0\le \theta \le 2\pi$ has a length of $L$ and is closed by a region that has an area $A$. How can I find the area of a region closed by polar curve say $r=4f(\theta)$ with same interval? How would I also find the length of say a polar curve $r=-3f(\theta)$ with same limits?

Answer 1

Hint:

Given $r=f(\theta)$ the area of the sector from the center $(0,0)$ and a arc of curve for $a\le\theta\le b$ is:

$$ A= \dfrac{1}{2}\int_a^b [f(\theta)]^2 d\theta $$ and the arc leght is:

$$ L=\int_a^b \sqrt{[f(\theta)]^2+\left(\dfrac {df(\theta)}{d\theta}\right)^2} d \theta $$

So you can find your answers.

For a proof and some exercise you can see here.