Daily Archives: March 12, 2009

A fun double integral

On my multivariable calculus class’s current problem set, I put a number of really challenging problems. One of them — from both the Exeter Math 5 course (here) and also in Anton — has students evaluate the following double integral, and then has students change the order of integration and then evaluate the double integral.

$\int_0^1 \int_0^1 \frac{x-y}{(x+y)^3}dydx$

Students expect the answers to be the same, but it turns out they are not. (Do you see why?)

Anyway, I have to say that I’m not a master integrator; it usually takes me a little longer than desired to figure out the best method to integrating. But I enjoyed the roads I took, so I thought I’d share the integral with you if you wanted a challenge.

And for those of you who know calculus, but forgot or never learned multivariable calculus, the problem reduces to you solving the following single integral: $\int_0^1 \frac{a-y}{(a+y)^3}dy$ , where $a$ is just a constant.