00:01
Okay, so for this equation here, we're looking at finding the volume contained between this paraboloid and the z equals 5 plane.
00:09
So we've got our paraboloid 13 minus 8 x squared minus 8 y squared.
00:16
So this looks like this.
00:17
It's an upside down paraboloid, so it's a hat -shaped with a maximum at z is 13.
00:26
And then it kind of drapes down intersecting the second plane at z equals 5, and then eventually intersecting the x -y -plane.
00:34
So we want to be working in polar coordinates here.
00:37
So we're going to transform from our x, y, z, cartesian coordinates into our r -theta -z cylindrical polar coordinates.
00:47
And to do that, we're just going to recognize that x equals r -cos theta and y equals our sine theta.
00:58
And so plugging those into this equation for z here, we can see that z becomes 13 minus 8 r squared and then we're going to get cos squared plus a sign squared turn and that's going to evaluate to 1.
01:13
So this is our new expression for z.
01:16
Thinking now about the volume, we know that the volume is going to be the integral between some limits of z are d r dr d theta.
01:32
And so we've just worked out what will go inside the integral...