Finding Moments for Elliptical Region

[City88]

Homework Statement

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

Homework Equations

Listed above...

The Attempt at a Solution

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.

 

[nicksauce]

Those boundaries describe a rectangle, not an ellipse. Do you see why?

 

[City88]

Ooohh yes, you're right.

So does this mean that I have to find the bounds of my integral as functions?
Would I have to find the equation of the ellipse in terms of y, and make that my bounds for the y-integral. and then keep the x integral as numbers?

like D = {(x,y)| a \leq x \leq b, g_{1}\leq y \leq g_{2}}

 

[nicksauce]

Yes that's what you would do. This problem may be easier in polar coordinates.

 

[City88]

Actually I just found out that I might have to use polar coordinates.
But since the region is not a circle, but an ellipse, how would I be able to write my x and y in polar coordinate form?
Normally we have x=rcos(theta), and y=rsin(theta) for a circle.

would it be...
x=4rcos(theta) + 2
y=6rsin(theta) + 4