Double integral in polar coordinate

[agro]

Homework Statement

With a > 0, b > 0, and D the area defined by

$$D: \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1$$

Change the integral expression below:

$$\iint\limits_D (x^2+y^2) dx\,dy$$

by using x = a r cos θ, y = b r sin θ. After that evaluate the integral.

The Attempt at a Solution

First of all, I can already visualize that the area is an ellipse, and because of that if we want to express the outside boundary in polar coordinates, it is

$$r_1(\theta) = \frac{ab}{\sqrt{(b \, cos \theta)^2+(a \, \sin \theta)^2}}$$

Therefore, since x^2 + y^2 = r^2, the integral can be expressed in terms of θ and r as:

$$\int_0^{2\pi} \int_0^{r_1(\theta)} \! (r^2)r \, dr\,d\theta$$

I've also understood how this construct comes from the Riemann sums.

However, I don't understand the substitution hinted from the problem. For every random point (r, θ), clearly x = r cos θ and not x = a r cos θ! Can someone give a geometric interpretation/explanation, and from that show how one changes the integral expression?

Thanks a lot.

 

[Raskolnikov]

x = r cos θ and y = r sin θ maps out a circle, not an ellipse.

The equation of an ellipse is:

x = a cos θ
y = b sin θ

Substitute these into your integral.