Triple Integral Question

**Redding1234** · April 4th 2010, 05:35 PM

I have the following problems:

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

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

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 $\theta$ , $\phi$ , and $\rho$ . Here, I am at a loss. Any thoughts about what I should use for the limits of each of them?

**Anonymous1** · April 4th 2010, 05:41 PM

Originally Posted by Redding1234

I have the following problems:

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

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

Can't you just do...

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

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

Continuing in the manner will give you your solution.

**Redding1234** · April 4th 2010, 05:50 PM

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.

**Anonymous1** · April 4th 2010, 05:52 PM

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 $[e^{-x^2}e^{-y^2}]\frac{1}{2}\sqrt{\pi}$ $erf(z)$

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