Evaluate $\iint_{D} \frac{y^2dxdy}{x}$

Question

$D$ is the region bounded by curves $y=x^2$,$y=4x^2$,$xy=1$,$xy=5$

Evaluate: $$\iint_{D} \frac{y^2dxdy}{x}$$

I found the four intersection points, but no clue how to proceed to choose limits of integration?

Answer 1

Let

• $u=xy \implies 1\le u \le 5$
• $v=y/x^2\implies 1\le v \le 4$

the Jacobian is

$$du\,dv=\begin{vmatrix}y&x\\-\frac{2y}{x^3}&\frac{1}{x^2}\end{vmatrix}dx\, dy=\frac{3y}{x^2}\,dx \,dy$$

therefore

$$\iint_{D} \frac{y^2dx\,dy}{x}=\frac13\int_1^5\int_1^4 xy\,du \,dv=\frac13\int_1^5\int_1^4 u\,du \,dv$$