00:01
For this problem, we are asked to set up a triple integral for the volume of the solid bounded by z equals the square root of 16 minus x squared minus y squared and z equals zero.
00:11
To begin, we'll note that if we set up our 3d axes or our 3d space here, x, y, z, then we note that we have that z is bounded below by, well, it's bounded below by z equals zero, so it's bounded below by the x, y, plane.
00:29
And we have the uppermost limit would be square root of 16 minus x squared minus y squared, which i'll note that if we do a little bit of a rearranging, we can see that that equation is equivalent to the upper half of the sphere described by x squared plus y squared plus z squared equals 16.
00:45
So we have that when x and y equals zero, we should have z equals eight, and then we should be having the upper half or the upper hemisphere of a sphere.
00:57
So it's just a little bit of a rough sketch there.
01:01
And we can see that the projection into the xy plane would be corresponding to x squared plus y squared equals 16.
01:11
So depending on how we want to order this, we can have a few different ways that we write out to everything...