Exercise 17

Let $D'$ be the region in the $xy$-plane that is the parallelogram with vertices $(3,3)$, $(4,5)$, $(5,4)$, and $(6,6)$.

1. Sketch and label the region $D'$ in the $xy$-plane.

2. Consider the integral $\iint_{D'} (x+y) \, dA$. Explain why this integral would be difficult to set up as an iterated integral.

3. Let a change of variables be given by $x = 2u + v$, $y = u + 2v$. Using substitution or elimination, solve this system of equations for $u$ and $v$ in terms of $x$ and $y$.

4. Use your work in (c) to find the pre-image, $D$, which lies in the $uv$-plane, of the originally given region $D'$, which lies in the $xy$-plane. For instance, what $uv$ point corresponds to $(3,3)$ in the $xy$-plane?

5. Use the change of variables in (c) and your other work to write a new iterated integral in $u$ and $v$ that is equivalent to the original $xy$ integral $\iint_{D'} (x+y) \, dA$.

6. Finally, evaluate the $uv$ integral, and write a sentence to explain why the change of variables made the integration easier.