How to evaluate $\int_A x^2 + y^2 d\lambda^2(x, y)$ with the change-of-variables formula?

Question

I would like to evaluate

$$\int_A x^2 + y^2 d\lambda^2(x, y)$$

with

$$A = \{x + y \le 2, x, y \ge 0\}$$

by applying the change of variables formula.

First things first, we know that

$$0 \le x + y \le 2.$$

In order to apply the change-of-varialbes formula, I need a proper substitution, and I think it would be beneficial to define

$$u = x^2,$$

$$v = y^2.$$

Hence,

$$0 \le \sqrt u + \sqrt v \le 2.$$

But where to go from here? I know that I need to end up with a double integral, but what are their boundaries then?

Answer 1

use $A=\{0<x<2,~~0<y<2-x\}$ $$\int_A x^2+y^2dxdy = \int_0^2 \left(\int_0^{2-x} x^2+y^2dy\right)dx$$