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

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

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?

$0\le u\le 4$ and $0\le v\le (2-\sqrt{u} )^2$ are boundaries for you changes of variable — 2017-02-11

user id:385707 17k · Accepted Answer

0

use $A=\{0

asked 2017-02-11

user id:385707

17k

1

Thanks, but I would like to do it with the help of the change-of-variables formula. – 2017-02-11
0

https://en.wikipedia.org/wiki/Integration_by_substitution#Substitution_for_multiple_variables – 2017-02-11
0

$0\le u\le 4$ and $0\le v\le (2-\sqrt{u} )^2$ are boundaries for you changes of variables – 2017-02-11

1 Answers 1