1. ## Triple Integral Question

I have the following problems:

(a) Evaluate $\displaystyle \int_{R^{3}} e^{{-(x^2+y^2+z^2)}}dV$

(b) Evaluate $\displaystyle \int_{R^{3}} e^{{-(x^2+2y^2+3z^2)}}dV$

I know how I am basically going to approach these (I am going to use spherical coordinates).

However, in other problems I was either given or could figure out the limits of $\displaystyle \theta$, $\displaystyle \phi$, and $\displaystyle \rho$. Here, I am at a loss. Any thoughts about what I should use for the limits of each of them?

2. Originally Posted by Redding1234
I have the following problems:

(a) Evaluate $\displaystyle \int_{R^{3}} e^{{-(x^2+y^2+z^2)}}dV$

(b) Evaluate $\displaystyle \int_{R^{3}} e^{{-(x^2+2y^2+3z^2)}}dV$
Can't you just do...

$\displaystyle \int_{R^{3}} e^{{-(x^2+y^2+z^2)}}dV = \int\int\int e^{-x^2}e^{-y^2}e^{-z^2}dzdydx = \int \int [e^{-x^2}e^{-y^2}]\int e^{-z^2}dzdydx$

Since $\displaystyle [e^{-x^2}e^{y^2}]$ can be treated as a constant when you are integrating $\displaystyle w.r.t.$ $\displaystyle z...$

Continuing in the manner will give you your solution.

3. Thanks for that approach! Do you know how the bounds of each variable should be handled? That's something I'm still not sure about.

4. I don't think it is a definite-triple integral. Just find the integral function, i.e., indefinite.

BTW have you ever heard of the error function? Because the solution to your first integration is $\displaystyle [e^{-x^2}e^{-y^2}]\frac{1}{2}\sqrt{\pi}$ $\displaystyle erf(z)$