00:01
In this problem, we want to evaluate the following triple integral using cylindrical coordinates.
00:05
So here we have the triple integral of x squared plus y squared over region e, where e is the region that lies inside the cylinder of radius 3 that is comprised between the planes z equal to minus 6 and z equal to 1.
00:23
Let's first start by drawing out the region we are interested in.
00:30
So first i will draw a normal 3d cartesian grid, and our cylinder here will be aligned with the z -axis and would theoretically be an infinite cylinder, but we are only interested in the portion of the cylinder between minus 6 and 1.
00:53
So our cylinder, actually i'll draw the cylinder in red, the cylinder will look something like this, an infinite tube, and in the xy plane it's going to look just like a circle with the radius of 3.
01:24
And we are interested in the portion of this tube between z equal to minus 6 to 1.
01:49
So essentially cutting off our cylinder like so.
01:58
So now we want to evaluate this triple integral using cylindrical coordinates.
02:01
So in cylindrical coordinates, i recall that we use the variables r, theta, and z, where r is defined as the square root of x squared plus y squared, and theta here is the angle made with respect to the x -axis.
02:36
For instance, i'll draw it in green, this angle right here will be theta, and this length here will be r, the radius.
02:50
And z remains the same as in cartesian coordinates.
02:54
And what we want to know is that the volume element in cylindrical coordinates is equal to r dr d theta dz...