00:01
So in this question, we want to find the average value of the function f of xy equals x squared plus y squared over the square having vertices at 0 0, 02, 22, and 200.
00:17
So how do we find the average value of a function over a region in the plane? the average value of a function of two variables is defined to be one over the area of that region times the double integral over that region of f of x, y, da.
00:48
So this time i have a square that's 2 by 2, and so the area of this region is 4.
00:57
This is times the double integral of our function, so the quantity of x squared plus y squared.
01:08
And since my region is a square, i don't think it matters if i set this up in the dx, d, y, d, y, d, d, y, direction.
01:19
And so for the sake of argument, let's set this up d, y, d x.
01:25
I'll have my y's range from zero to two, and my x is.
01:31
May range from 0 to 2 as well.
01:37
So now, evaluating this double integral, i'm first going to need an anti -derivative with respect to y.
01:47
My anti -derivative of x squared with respect to y is x -squared y.
01:54
My anti -derivative of y squared with respect to y, y -cubed over 3.
02:03
And this is evaluated from 0 to 2, dx.
02:13
And so now we'll evaluate.
02:14
We'll say, okay, i've got a quarter times the integral from zero to two...