Hey guys I am having difficulties in problem 5. I thought I understood it, but I suppose I was mistaken. I will now explain what I planned to do to solve this problem and where I got stuck.
So I thought that the solution to the problem would lend itself the following way. I was to calculate $I$ for both these circles individually and then take the difference of the larger $I_1$ circle from the smaller $I_2$. So I let $I=I_1 - I_2$. However, when I began to calculate $I_1$ $I_1$$=0$, but for some reason this didn't seem quite right to me. I will show you my work. Please let me know if my methodology is wrong or my calculations. Thank you in advance :)
$Q(x,y) = x^4 +2x^2 y^2$ $\implies$ $\frac{\partial Q}{\partial x} = 4x^3 +4xy^2$ $\wedge$ $P(x,y) = 4xe^{-4x} \implies \frac{\partial P}{\partial y} = 0$
Now from here I obtain $I_1 = \iint\limits_D 4x^3 + 4xy^2 dA$, where $D=\{(x,y)|x^2 + y^2 = 9\}$ by Green's Theorem
However, I wish to evaluate this integral using polar coordinates. Thus, let $x = rcos\theta$, $y=rsin\theta$, and $dA = rdrd\theta$ with $R=\{(r,\theta)| 0\le r\le3,0\le \theta\le2\pi\}$. Thus, $I_1 = \int\limits_0^3 \int\limits_0^{2\pi} 4r^4cos\theta drd\theta = 0$.
