4
$\begingroup$

I have the following to polar coördinates: $r=1+\cos(\theta)$ and $r=3\cos(\theta)$.

The question is to calculate the area in side $r=1+\cos(\theta)$ and outside $r=3\cos(\theta)$. I know I need to use the formula $$\int\frac{1}{2}r^2d\theta$$ But I don't really know which boundaries to choose for both polar coördinates.

$\endgroup$
  • $\begingroup$ The tag says "polar-coordinates" (not polar coördinates:) ) $\endgroup$ – Dietrich Burde Jun 5 at 18:51
1
$\begingroup$

Sometimes one function is outside, sometimes the other. How do you want to integrate such a region?

enter image description here

$\endgroup$
1
$\begingroup$

When you draw a graph you will notice that $r=1+\cos\theta$ is a cardioid and $r=3\cos\theta$ is a circle.

enter image description here

To find the area inside $r=1+\cos\theta$ and outside $r=3\cos\theta$ you need to split the integral into two parts.

You also need to find out the intersection point of those two areas.

So, to find that we need $$1+r\cos\theta=3\cos\theta$$ $$\theta=\dfrac{\pi}{3},\dfrac{5\pi}{3}$$ So, the Area we need is $$A=\int_{\frac{\pi}3}^{\frac{\pi}{2}}\int_{3\cos\theta}^{1+\cos\theta}r\ drd\theta=1-\dfrac{\pi}{4} $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.