Improper integral at unbound domain

Question

I'm trying to calculate the improper integral $$\int\int_D\frac{1}{x^4+y^2}dxdy$$ where the domain is : $$D = ((x,y)|x\ge1,y\ge{x^2}) $$ I'm struck at figuring out how to change the parameters adequately, and determining the new domain. Any ideas?

Answer 1

Let $\dfrac{y}{x^2}=u$ and $x=v$ then $1\leq u<\infty$ and $1\leq v<\infty$, also $|J|=-v^2$, we find $$\int\int_D\frac{1}{x^4+y^2}dxdy=\int_1^\infty\int_1^\infty\frac{1}{1+u^2}\frac{-1}{v^2}dudv=\color{blue}{\dfrac{\pi}{4}}$$

Answer 2

Hint: You may use a change of coordinates $(x,y)=(x,x^2 t)$ after which the integral separates and is easily calculated (if I didn't make a mistake)