00:01
Right, so we're supposed to evaluate this integral over this volume.
00:03
I am going to use cylindrical coordinates, okay, so that z stays the same, but we know that rho squared is x squared plus y squared, and then, of course, we usually think of that as like this, and that way.
00:41
Okay so this so our integral is going to look like this we're going to integrate on rho from zero to the square root of six that's d rho okay and then we're going to integrate on phi from zero to two pi of course we actually do the integrations in the opposite order so then with z z we integrate from that's 8 plus rho squared in the upper limit okay and we also have a factor of rho from our volume element and then we got e to the z all right i'll bring that row over here and then our phi integral is always going to give us 2 pi and then this integral gives us z, excuse me, he gives us e to the z, and we evaluate that from 0 up to 8 plus rho squared.
02:24
Bring the 2 pi out in front, and we got this rho d rho, and then we got our upper limit is e to the 8 plus rho squared, and then we got e to the 0, which is 1.
02:52
And i'm going to kind of rewrite this a little bit.
02:59
And then this term, we factor out the e to the 8.
03:08
All right.
03:11
So this is a sum of two integrals.
03:14
The second one is easy to do.
03:16
In fact, the first one is also easy to do.
03:19
We make, we'll assign u is rho squared.
03:25
And then du is 2 rho d rho.
03:30
Okay.
03:31
Okay, so this integral becomes, so when rho is square root of 6, u is equal to 6, so it's 0 to 6.
03:47
We got a 1 half of du, and then e to the u, and then out in front, we get that e to the 8...