Change of Variables — Question 4

Question 4 – Solution

Strategy. Each pair of parallel boundary lines becomes a pair of constant-coordinate sides.

Step 1: Geometry

The image is the rectangle $0\le u\le 2$, $1\le v\le 4$. The coefficient matrix has determinant $$\frac{\partial(u,v)}{\partial(x,y)} =\begin{vmatrix}1&2\\3&-1\end{vmatrix}=-7.$$ Since this determinant is nonzero, the linear map is globally one-to-one.

Step 2: Change variables We have $dA=\frac 17\,du\,dv$ and the integrand is $uv$. Hence $$\begin{align*} I&=\frac 17\int_0^2\int_1^4uv\,dv\,du\\ &=\frac 17\left[\frac{u^2}{2}\right]_0^2 \left[\frac{v^2}{2}\right]_1^4 =\boxed{\frac{15}{7}}. \end{align*}$$

Verification On the transformed rectangle, $u\ge 0$ and $v>0$, so the positive answer has the required sign. The constant area scale $1/7$ also matches the inverse determinant.