00:01
Okay, so this problem wants you to evaluate the triple integral of x square over the region of e.
00:07
Or e is bounded by a cylinder of x squared plus y squared equals to one, a horizontal plane of z equals 0, and a cone with the formula of z square equals to 4 times x square plus y square.
00:21
Okay, so the first thing we need to do is to convert everything into cylindrical coordinates.
00:28
So if we were to do that, we would have our triple integral integral.
00:31
Of x squared and x squared in cylindrical coordinates is r squared times cosine of square theta because x is our cosine data in cylindrical coordinates.
00:45
And if we would convert dv into cylindrical coordinates, it would be r times dv.
00:51
It would be r times d z, dr, d theta.
00:55
So we can just replace the squared with a cubed and just add a dz, d r d3d3 .3.
01:03
So now looking at our boundaries, first we have the cylinder of radius 1, and we know that x squared plus y squared equals to r square.
01:14
So we can rewrite this as r equals to 1.
01:18
The plane, the horizontal plane z equals 0, well, that's just simply z equals 0 in cylindrical coordinates.
01:24
And our cone of z squared equals to 4x2 plus 4 y squared, well, we know that x with y squared equals to r squared.
01:32
So we can rewrite this cone as z equals to 2r.
01:38
And as you can see on the right, i have the plot of our boundaries.
01:43
The horizontal plane of z equals zero is essentially the xy plane.
01:49
And we have the red cylindrical cylinder of xxxxx2 equals the 1 or r equals to 1.
01:54
And the blue region is the cone.
01:58
And the region we're integrating is this part right here.
02:03
Where they overlap.
02:05
All right, so let's figure out our boundaries.
02:08
So first is dz, and we know that dz starts at the, for the interval for z, we know that it starts at zero because of the horizontal plane, and we know that the cone is the upper boundary, and that is 2r...