Compute the integral, $\iint_R x^2+xy^3 dA$ where R is bound $0\leq y \leq 2$

Question

Compute the integral, $\iint_R x^2+xy^3 dA$ where R is bound $0\leq y \leq 2$

I found this question on one of my lecutre tutorials. I just want to know if its possible to calculate this without knowing the range for x? I highly doubt that this is a misprint. I would like to get some ideas?

P.S. I know how to do the double integral if a range for x is given.

Answer 1

The way this is specified, one assumes $R$ is a vertical strip with $x$ anywhere in $\mathbb{R}$, so you get $$ \int_{-\infty}^\infty \int_0^2 \left(x^2 + xy^3\right)dy\ dx = \int_{-\infty}^\infty \left(2x^2 + 16x\right) dx, $$ which diverges. Most likely, this is a misprint, with intent to write $0 \le x,y \le 2$, which would converge...