Find the average value of the function f(x,y)=x^2+y^2

[een]

Homework Statement

Let a>0 be a constant. Find the average value of the function f(x,y)=x^2+y^2
1) on the square -a$$\leq$$x$$\leq$$a, -a$$\leq$$y$$\leq$$a
2) on the disk x^2+y^2$$\leq$$a^2

Homework Equations

The Attempt at a Solution

1) I integrated $$\int$$a-(-a) $$\int$$a-(-a) (x^2+y^2) dxdy and got (8/3)a^4..Is this right?

2)I converted it to polar coordinates 0$$\leq$$$$\theta$$$$\leq$$2pi
and 0$$\leq$$r$$\leq$$sqrt(a)
i integrated $$\int$$0-2pi$$\int$$0-sqrt(a) r^2drd$$\theta$$
and got 2/3pi*(sqrt(a)^3)... is this right?----- 2pi$$\frac{\sqrt{a}^3}{3}$$

 

[Mark44]

For 1, I get (2/3)a^2 for the average value. Because of the symmetry of the region and the integrand, I took a short cut and integrated from 0 to a for both x and y, and multiplied the result by 4. Don't forget that for the average value, you have to divide by the area of the region, which is 4a^2. Your answer divided by 4a^2 equals mine.