Using a change of variables, the double integral $\iint$_{R $( x + y ) e ^ { x - y } d A$}where R is the region bounded by the quadrilateral with vertices (4, 0), (6, 2), (4, 4), and (2, 2) is A) $12 \left( e ^ { 2 } - 1 \right)$ B) $- 12 \left( e ^ { 4 } + 1 \right)$ C) $- 12 \left( e ^ { 4 } - 1 \right)$ D) $12 \left( e ^ { 4 } - 1 \right)$ E) $12 \left( e ^ { 4 } + 1 \right)$

Question

Quizplus · Accepted Answer

The correct choice is to use a change of variables that simplifies the integral. By setting $u = x + y$ and $v = x - y$, the region of integration becomes simpler, and the Jacobian of the transformation helps in evaluating the integral correctly. The given integral, after transformation and evaluation, yields $12(e^4 - 1)$, making choice D correct.

Using a Change of Variables, the Double Integral ∬\iint∬ R (x+y)ex−ydA( x + y ) e ^ { x - y } d A(x+y)ex−ydA

Using a Change of Variables, the Double Integral $\iint$ _{R $( x + y ) e ^ { x - y } d A$}