00:01
So in this question, we want to evaluate our integral by changing to spherical coordinates.
00:06
So first of all, what do i know about my integrand, x squared plus y squared plus z squared? well, we know that in spherical coordinates, that that is just row square.
00:17
That's one of my formulas, that x squared plus y squared plus z squared is equal to row squared.
00:23
My differential, dz, d, y, dx, i know that changes to row squared, sine of phi, d row, d theta, d, phi, when i go over to spherical coordinates.
00:42
Now we think about our limits of integration.
00:46
So my z goes from zero to the square root of x squared plus y squared plus z squared, plus z squared.
00:56
So we've got z equals 0 and z equals the squared of x squared, or excuse me, and we've got, yes, z equals the square of x squared plus y squared to convert into spherical coordinates.
01:16
Now z equals zero.
01:18
You'll remember that z is equal to row cosine of phi, row cosine of phi equals zero.
01:25
And that means the cosine of phi is zero, that happens at phi equals pi over two.
01:34
Now, if z equals the square root of x squared plus y squared, you know that the squared of x squared plus y squared, that that's r.
01:45
You know that z again is row cosine of phi, while r is row sine of phi in spherical coordinates.
01:57
This implies that cosine of phi, equals the sine of phi, and that happens when phi equals pi over four.
02:09
So, phi equals pi over four.
02:11
Now, thinking about my row and my theta, well, when we projected onto the x, y, plane, my y's, they were extending from negative the square root of one minus x squared to positive the square.
02:35
Root of 1 minus x squared.
02:40
What does that look like? well, if you have y equals plus or minus the squared of 1 minus x squared, if you square both sides, you get y squared equals 1 minus x squared, you get x squared plus y squared equals 1.
02:57
You're getting the unit circle in the x y plane.
03:01
The y equals negative the square root of 1 minus x squared that's the bottom half of the unit circle while y equals positive the squared of 1 minus x squared that's the top half and my x is they're going from negative 1 to 1 and so i'm getting this entire unit circle that means that i'm going all the way around in terms of theta that my theta's they're going from 0 to 2 to 2.
03:37
To p.
03:39
And notice, i'm going to have here a sphere of radius 1, as my xes are going from negative 1 to 1 over top of this region that, again, is a circle of radius 1 when projected into the x, y, plane.
03:59
And so my row this time, that'll range from 0 to 1...