Marginal Density of Coordinates Inside an Ellipse

[gajohnson]

Homework Statement

A point is chosen randomly in the interior of an ellipse:

(x/a)^2 + (y/b)^2 = 1

Find the marginal densities of the X and Y coordinates of the points.

Homework Equations

NA

The Attempt at a Solution

So this ought to be uniformly distributed, thus the density function for (x,y) is f_{x,y} = 1/∏ab (where ∏ab is the area of the ellipse)

So, to find the marginal density for x (and later for y), I realize that I just need to find the limits of integration and then go about my business. I believe that the limits of integration are
-(b/a)\sqrt{a^2 - x^2} and b/a\sqrt{a^2 - x^2},

since these should be the minimum and maximum values that y can take for any given x. Are these limits of integration and/or is my reasoning correct?

As always, many thanks to all of you wonderful Homework Helpers!

 

[clamtrox]

That seems entirely correct.

 

[gajohnson]

That seems entirely correct.

Excellent, thanks!