Let $A$ be the set in $\mathbb{R}^2$ defined by $A = \left\{(x,y)\left| x \gt 1\text{ and }0\lt y\lt\frac{1}{x}\right.\right\}.$ Calculate $\int\!\!\int_A\left(\frac{1}{x}\right)y^{1/2}dA$ if it exists.
*Important: There's only 1 integral sub A, this is not a double integral.
My proof:
So our integral will have bounds of $x$ from $1$ to $\infty$ and $y$ will have bounds from $0$ to $\frac{1}{x}$.
So, we have an integral of $1$ to $\infty$ of $1/x\; dx$ and an integral of $1$ to $1/x$ of $y^{1/2}\; dy$
and we have an integral of $1$ to $\infty$ of $\left(\frac{2}{3}\left(\frac{1}{x}\right)^{3/2}-\frac{2}{3}\right)\frac{1}{x}\,dx$ and our final answer diverges so it doesn't exist.
But how to say this rigorously/correctly?
Thanks