Change of Variables — Question 5

Question 5 – Solution

Strategy. The ratio $v=y/x$ turns the two sloping sides into constant-coordinate boundaries.

Step 1: Inverse and bounds Since $x>0$ on $R$, $$x=u,\qquad y=uv.$$ The region becomes the rectangle $$1\le u\le 3,\qquad 1\le v\le 2.$$ The inverse map is one-to-one because $u=x$ and $v=y/x$ recover each point uniquely.

Step 2: Jacobian $$\frac{\partial(x,y)}{\partial(u,v)} =\begin{vmatrix}1&0\\v&u\end{vmatrix}=u.$$ Thus $$I=\int_1^3\int_1^2\frac 1u(u)\,dv\,du =\int_1^3\int_1^2 1\,dv\,du=\boxed{2}.$$

Verification Direct Cartesian integration gives $$\int_1^3\int_x^{2x}\frac 1x\,dy\,dx =\int_1^3\frac{2x-x}{x}\,dx=2.$$ The two methods agree exactly, including the variable area scale $u$.