By introducing new variables u and v in place of x and y
By introducing new variables $u$ and $v$ in place of $x$ and $y$, determine the limits of integration for: $$ \int_0^2 dx \int_{1-x}^{2-x} z(x,y)\,dy, $$ using the substitution $u=x+y$, $v=x-y$.
Explanation of Method
The two slanted boundaries $y=1-x$ and $y=2-x$ are both lines of slope $-1$, so each becomes a constant value of $u=x+y$. The two vertical boundaries $x=0$ and $x=2$, however, are not aligned with either new coordinate axis, so they become slanted lines in the $(u,v)$-plane, producing a parallelogram (not a rectangle) as the transformed region.
- Transform each of the four boundaries into $(u,v)$-coordinates.
- Compute the Jacobian of the inverse transformation.
- Write the transformed integral over the resulting parallelogram.
Step 1 · Transform the boundaries
Solving $u=x+y$, $v=x-y$ gives $x=\dfrac{u+v}{2}$, $y=\dfrac{u-v}{2}$.
$$ y=1-x \Rightarrow x+y=1 \Rightarrow u=1, \qquad y=2-x \Rightarrow x+y=2 \Rightarrow u=2. $$
$$ x=0 \Rightarrow \frac{u+v}{2}=0 \Rightarrow v=-u, \qquad x=2 \Rightarrow \frac{u+v}{2}=2 \Rightarrow v=4-u. $$
So $u$ runs from $1$ to $2$, and for each $u$, $v$ runs from $-u$ to $4-u$: the region $D'$ is a parallelogram (a rectangle in the original $x$-strip becomes a slanted parallelogram in $u,v$, since $v$'s bounds shift linearly with $u$).
Step 2 · Compute the Jacobian
With $x=\dfrac{u+v}{2}$, $y=\dfrac{u-v}{2}$:
$$ \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix}\frac12&\frac12\\[2pt]\frac12&-\frac12\end{vmatrix} = -\frac12, \qquad dx\,dy = \frac12\,du\,dv. $$
Step 3 · Write the transformed integral
$$ \int_0^2 dx\int_{1-x}^{2-x} z(x,y)\,dy = \int_1^2\int_{-u}^{4-u} \frac12\,z\!\left(\frac{u+v}{2},\frac{u-v}{2}\right)dv\,du. $$
Additional Notes
- Unlike a substitution aligned with the region's own boundaries (which produces a rectangle), this substitution is aligned with only two of the four sides, so the transformed region remains a parallelogram — the limits on $v$ still depend on $u$.
Result
The limits of integration and transformed integral are: