Keep reading to see how I solved this problem.

Did you come up with another solution? Tell us in the comments, and good luck on whatever tests you may encounter!

As finals for many an institution loom closer and closer, here's an Advanced Knowledge Problem that might very well be found on a Multivariable Calculus exam. If it looks too hard, try looking up and contemplating the method that's used to calculate the "usual" Gaussian integral, that is, the integral below without the r-squared term.

Keep reading to see how I solved this problem.

Did you come up with another solution? Tell us in the comments, and good luck on whatever tests you may encounter!

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I used partial integration, with f=r and g'=r*e^(-r^2). Seems way more easy to me?

ReplyDeleteGood point, Judith! There's often more than one way to solve any problem, and sometimes I can miss what might seem obvious to you.

DeleteWhen solving standard Gaussians I always try to avoid iterated integrals. IBP is quite a bit faster. Also, and i didn't work it out all the way, but another trick is to square the non-standard Gaussian to get the same result, again only needing to evaluate one integral.

DeleteJust some tricks from the physics side of things.

The integral is essentially the second moment of a Gaussian random variable with mu =0 and sigma^2 = 1/2. Then the result is immediate, since we have Second moment = sigma^2 = 2/sqrt(pi) {problem integral}

ReplyDeleteI also split the integral into -0.5 * (integral r * -2*r*e^(-r^2)) and solved that by partial integration to -0.5 * (r*e^(-r^2) - integral e^(-r^2)) with boundaries of 0 to infinity. By knowing that the integral e^(-r^2) from 0 to infinity is sqrt(pi)/2 and by solving r*e^(-r^2) for infinity to be 0 and r*e^(-r^2) for 0 to be 0 too, it simplifies to -0.5 * (0 - (-sqrt(pi)/2)), which is sqrt(pi)/4

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