Calculate the following integral in the rectangle $[0,1]$x$[2,2]$

Question

I have to calculate the following integral: $\int_{0}^{1} \int_{2}^{2} arctan(xy^{31}) dx dy$

I've done double integrals in the past but I've never been in a situation where I can not separate the two variables so I don't know where to begin. Any words of help will be great to start.

Answer 1

Take a look at the limits of the integral, as you are integranting in the interval $[2,2]$ the integral vanish, $\int_{0}^{1}dx(\int_{2}^{2}dy \; acrtan(x y^{31}))=0$. See Fubini´s theorem to learn about the integral of non-separable functions.