# 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.

Have fun. And for what the double integrals turn out to equal, go below the jump.

The double integrals turn out to be 1/2 and -1/2, depending on the order of integration. The reason they aren’t the same is slightly complex and involves looking at the graph of the function, but the important thing to note is that there is something crazy going on at the point (0,0).

1. Daniel says:

I’m very late to the party, but I can’t see how that integral converges. If you do partial fractions decomposition and integrate \int_0^1 \frac{a-y}{(a+y)^3}dy, the resulting integral cannot be integrated from 0 to 1.

1. Daniel says:

Oh, I found a mistake in my calculations. Sorry!