# Thread: Triple integral over the sphere centered at (1,1,1) and radius 2

1. ## Triple integral over the sphere centered at (1,1,1) and radius 2

Problem:

Let f(x,y,z)= xy+yz. Evaluate the triple integral of the function over the sphere centered at (1,1,1) and radius 2.

Attempted solutions:

I'm having trouble visualizing what this question is asking. So far I've done this:

1. Decided I'm probably going to use spherical coordinates given I'm working with a sphere.

2. Converted the integral to spherical coordinates.

Questions:

1. How do I go about describing the given sphere in spherical coordinates? I know how to describe the unit sphere, but I can't figure out how to describe spheres that have been transformed (such as this one).

2. Am I finding the volume beneath xy+yz bounded below by the sphere?

Any other tips would be great! Thanks.

2. First "beneath xy+ yz" makes no sense. In order to have a surface that must be equal to something. What is it equal to?

The equation of the sphere with center (1, 1, 1) with radius 2 is $\displaystyle (x- 1)^2+ (y- 1)^2+ (z- 1)^2= 4$.

In spherical coordinates, $\displaystyle x= \rho cos(\theta)sin(\phi)$, $\displaystyle y= \rho sin(\theta)sin(\phi)$, $\displaystyle z= \rho cos(\phi)$. Put those into the equation of the sphere and plane.

However, you might find it simpler to shift the entire problem so that the center of the sphere is at the origin. That is, let u= x- 1, v= y- 1, w= z- 1 so that in the uvw- coordinate system the sphere is given by $\displaystyle u^2+ v^2+ w^2= 4$ and the surface xy+ yz= f(x,y,z) becomes $\displaystyle (u+1)(v+1)+ (v+1)(w+1)= f(u+1,v+1,w+1)$ (here f is the missing part of the equation of the surface).
Now replace u, v, and w with $\displaystyle \rho cos(\theta)sin(\phi)$, $\displaystyle \rho sin(\theta)sin(\phi)$, and $\displaystyle \rho cos(\phi)$.

(Depending on exactly how the lower surface is defined, you might find it simpler to use cylindrical coordinates.)

3. "First "beneath xy+ yz" makes no sense. In order to have a surface that must be equal to something. What is it equal to?"

In the context of the problem it would be equal to "W" right? EDIT: ah you mean equal to a constant. nvm.

Thanks for the tips I will try them