Double integral that seems to go over undefined region

Question

Evaluate $\int_R {8\over7}x^2y^{−3}dydx$ where R = {$(x, y) : 1 ≤ x, y ≤ 2, x ≥ y$}.

I believe that the integral should be $$\int_1^{3}\int_0^{x} {8\over7}x^2y^{−3}dydx $$

However when I put this into a calculator, it says "undefined", obviously because of putting $0$ into the $y^{-3}$ of the inner integral. When I do it myself with limits it doesn't help as the inner integral becomes $ \infty $ (and the correct solution to the entire double integral is $.7619$)

What am I getting wrong here?

Answer 1

Since $1\le x,y \le 2$ and $y\le x$ the integral is

$$\int_1^{2}\int_1^x {8\over7}x^2y^{−3}dydx.$$

What am I getting wrong here?

Your limits of integration are wrong.

$\int_1^{3}\int_0^{x} {8\over7}x^2y^{−3}dydx$

You are assuming $0\le y\le x$ and $1\le x\le 3.$