Change of Variables — Question 2

Question 2 – Solution

Strategy. Product and ratio coordinates straighten the hyperbolas and rays into a rectangle.

Step 1: Region and inverse In the first quadrant, $$x=\sqrt{uv},\qquad y=\sqrt{\frac{u}{v}},$$ so positive $(u,v)$ determine a unique $(x,y)$. The image is $1\le u\le 4$, $1\le v\le 9$.

Step 2: Jacobian Differentiating the inverse formulas gives $$\left|\frac{\partial(x,y)}{\partial(u,v)}\right| =\frac{1}{2v}.$$ Therefore the factor $x/y=v$ cancels the $v$ in the denominator.

Step 3: Evaluate $$I=\int_1^4\int_1^9v\left(\frac 1{2v}\right)dv\,du =\frac 12(3)(8)=\boxed{12}.$$

Verification The region area is $\int_1^4\int_1^9(2v)^{-1}dv\,du=3\ln 3$. Since $1\le x/y\le 9$, the bounds $3\ln 3\le I\le 27\ln 3$ hold; $I=12$ lies between them.