00:01
In this problem, we want to evaluate the following triple integral.
00:05
That is the volume integral of x plus y minus 3z over region e.
00:14
Region e corresponds to the points x, y, z such that y ranges between minus 5 to 0, x ranges between 0 and y, and z ranges between 0 and x plus y squared.
00:27
So because our upper bound on z depends on x and y, we will integrate with respect to z first, then x, because x has an upper bound that depends on y, and then y.
00:48
So dv, which typically corresponds to products of dx, dy, dz in no particular order, we'll write as dz times dx times dy, where z will range between 0 and x plus y squared, x will range between 0 and y, and y will range between minus 5 and 0.
01:24
So let's integrate with respect to z first, and we obtain xz, yz minus 3z squared over 2, where z will range between 0 and x squared, oh no, x plus y squared...