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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

y = -x

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 =(

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    Lines perpendicular to $y = -x$ will be of the form $y = x + b$2012-03-08
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    See [here](http://www.intmath.com/plane-analytic-geometry/perpendicular-distance-point-line.php). This gives you the function you want to find the average value of over your region. You friend has the proper function, but you need to divide the integral you have by the area of the region.2012-03-08
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    The area is just a quarter of a circle right $\pi a^2/4$ ? =)2012-03-08
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    You may recall from a Linear Algebra course, or you can derive directly, that the distance from the point $(s,t)$ to the line with equation $ax+by+c=0$ is $\dfrac{|as+bt+c|}{\sqrt{a^2+b^2}}$.2012-03-08
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    Yes, that's correct.2012-03-08
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    By the way, if you're studying vector methods, in particular, projecting a vector onto another vector, you should have a formula in hand to give you the distance you want.2012-03-08
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    Also perhaps of interest is the fact that the answer will be the distance from the centroid of your quarter-disk to your line (which, by symmetry, will be the distance from the centroid to the origin). The calculation of the $x$ or $y$ coordinate of the centroid involves an easy one-variable integral.2012-03-08
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    @Robert Israel: Nice comment! One might add that if we are not in an integrating mood, we can find the centroid by the theorem of Pappus.2012-03-09

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