Question

Can I get some help with this polar coordinate question (Calc II)?

Answer 1

I have no idea where to even start with this one

Answer 2

I found the answer for part 1, in that I just need to sketch a "donut" with a radii 1 to 2, and fill in the region from pi/4 to pi

Answer 3

But how would I go about part b?

Answer 4

Here are some hints:

For (a), the equation \(\large r = a\,\,(a>0)\) in polar coordinates is equivalent to \(\large x^2+y^2=r^2\) in rectangular coordinates.

For (b), the sketch of the region is in the attacked figure. So if you want the area of the right half, you can compute the area of the entire region and then divide it in half, since the cardioid is symmetric about the y axis. The area enclosed by a polar graph is given by the formula
\[\large A = \frac{1}{2}\int_a^b [r(\theta)]^2\,d\theta\]
Hence, the area of the entire thing is
\[\large \frac{1}{2}\int_0^{2\pi}(1+\sin\theta)^2\,d\theta\]
and thus the area of the right half is
\[\large \frac{1}{2}\left(\frac{1}{2}\int_0^{2\pi}(1+\sin\theta)^2\,d\theta\right)=\frac{1}{4}\int_0^{2\pi}(1+\sin\theta)^2\,d\theta\]
Does this make sense? :-)

Answer 5

Forgot to attach the figure... >_>

Answer 6

Yes, that make a LOT of sense!

Thank you for helping. My instructor BLEW through the polar coordinate chapter, and nobody picked it up very well.

The polar area integral is familiar, but it's nice to have it explained.

Thank you very much @@ChristopherToni !