Interested in finding volumes with multivariables to understand the background

[adradmin]

I can't find the volume of solid sqrt(x) + sqrt(y) + sqrt(z) = 1. It's a graph but I wish I had a graphing calculator to see it. It's bounded by x=0, y=0, z=0. I'm teaching myself this stuff and think integration using a change of variables by making x=u^2? This would be a transformation of T and not T-1 right? I wonder if you guys know about these kinds of problems thanks

 

[HallsofIvy]

You have three variables, why just change x? To answer your general question, yes, you can change variables. If, for example, you have x, y, and z all functions of the new variables u, v, and w, you will need to change dxdydz, the "differential of volume" to the corresponding differential of volume in u, v, and w. Since u, v, and w will measure distances differently, of course, you can't expect it to be just dudvdw. In fact, you need to multiply by the "Jacobian". That is the determinant of the 3 by 3 matrix whose first row is the partial derivatives of u, with respect to x, y, and z, second row is the partial derivatives of v, with respect to x, y, and z, and third row is the partial derivatives of w, with respect to x, y, and z. I think that matrix is the "T" you are referring to.