I need to solve the following problem, given in my book.
Calculus A complete course 9th - Robert A Adams
14.4 25.
"Find the average distance from points in the quarter disk $ x^2 + y^2 \leq a^2 \ , \, x \geq 0 \ , \ y \geq 0, \,$ to the line $x + y = 0$.
I tried drawing an image as shown below
My friend says that the solution can be found by solving
$$ \large \int_0^a \int_0^\sqrt{a^2-x^2} \frac{x+y}{\sqrt{2}} \,\mathrm{d}x \, \mathrm{d}y $$
But I can not really see why this double integral works, it looks like we are always integrating the distance from $0$ to $a$. But in my eyes the distance changes.
I guess I need to find a line perpendicular to $y = -x$ and find the distance, but could anyone help me out? I've been sitting a few hours with this problem now =(