Calculating Moments: Finding M_{x} & M_{y} in Ellipse Bounded Region

[City88]

Hi there...
I'm not sure how to go about finding the following moments:

<br /> M_{x}= \int \int\ y dx dy
<br /> M_{y}= \int \int\ x dx dy
Where the region is bounded by the ellipse:
\frac{(x-2)^2}{16}} + \frac{(y-4)^2}{36}} = 1

I tried this several ways. I drew the ellipse and found the bounds to be
-2 \leq y \leq10
-2 \leq x \leq 6

Then I tried integrating with those bounds, but I can't seem to get the right answers. Any help/hints would be greatly appreciated.

 

[maze]

Consider changing variables to center the ellipse around zero. eg:
u = x - 2
v = y - 4