# Variable substitution problem

1. Nov 10, 2008

1. The problem statement, all variables and given/known data
$$\int\int\int _E\(x^2y}\;dV$$

Where E is the solid bounded by $$x^2/a^2+y^2/b^2+z^2/c^2=1$$

2. Relevant equations

variable substitution x=au, y=bv, z=cw.

3. The attempt at a solution

I found the jacobian (abc) and I substituted my variables but I can't find the limits of integration. The only equation I have for the limits is $$u^2+v^2+w^2\leq1$$. I don't know how to find the limits of integration for u, v, and w individually.

2. Nov 10, 2008

### Dick

Now switch to spherical coordinates in u,v,w.

3. Nov 10, 2008

### HallsofIvy

Staff Emeritus
First decide on the order in which you want do integrate:
$$\int dudvdw$$?

Fine. Project the figure on to the vw plane: v2+ w2= 1. Then project that onto the w line: the segment from w= -1 to 1. The limits on the outer "dw" integral have to be numbers. In order to cover the entire figure, w must vary from -1 to 1. For each w, then v must vary from $-\sqrt{1- w^2}$ to $\sqrt{1- w^2}$. Finally, for each v and w, u varies from $-\sqrt{1- v^2- w^2}$ to $\sqrt{1- v^2- w^2}$. Those are the limits of integration.

Of course, because u2+ v2+ w2= 1 is a sphere in uvw-space, spherical coordinates, as Dick suggested, are simplest. The limits of integration would be exactly the same as if it were x2+ y2+ z2= 1.

4. Nov 10, 2008