Calculating Polar Moment of a Region Inside/Outside Circle and Cardiod

[VinnyCee]

Here is the problem:

Find the polar moment of the region that lies inside the circle $$r = 3$$ and outside the cardiod $$r = 2 + \sin\theta$$. Assume $$\delta = r\theta$$

Here is what I have:

$$I_{0} = I_{x} + I_{y}$$

$$I_{0} = \int_{0}^{2\pi}\int_{2 + \sin\theta}^{3}\;r^3\;\theta\;\sin^2\theta\;dr\;d\theta + \int_{0}^{2\pi}\int_{2 + \sin\theta}^{3}\;r^3\;\theta\;\cos^2\theta\;dr\;d\theta$$

$$I_{0} = \int_{0}^{2\pi}\int_{2 + \sin\theta}^{3}\;r^3\;\theta\;dr\;d\theta$$

Is this the correct setup? I don't have to manually evaluate this one, I just need to setup the integral limits and the integrand. Thank you in advance!

 

[dextercioby]

It looks okay to me...The cardioide & the circle have only one common point (y=3)

Daniel.

 

[thepatientmental]

Yes, your setup looks correct. The integral limits and integrand are the most important parts when setting up a polar moment integral. As long as those are correct, you should be able to evaluate the integral using a calculator or computer program. Good job!