Using a change of variables, the double integral $\iint$_{R $( x + y ) ^ { 2 } \sin ^ { 2 } ( x - y ) d A$}where R is region bounded by the quadrilateral with vertices $( \pi , 0 ) , \left( \frac { 3 \pi } { 2 } , \frac { \pi } { 2 } \right) , ( \pi , \pi )$ and $\left( \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ is A) $- \frac { 7 \pi ^ { 4 } } { 12 }$ B) $\frac { 7 \pi ^ { 4 } } { 12 }$ C) $\frac { 5 \pi ^ { 4 } } { 12 }$ D) $- \frac { 5 \pi ^ { 4 } } { 12 }$ E) $- \frac { 11 \pi ^ { 4 } } { 12 }$

Question

Quizplus · Accepted Answer

To simplify the integral, we can use a change of variables. Let $u = x + y$ and $v = x - y$. The Jacobian of this transformation is $J = \frac{1}{2}$. The region R in the uv-plane becomes a rectangle, and the integral simplifies to $\frac{1}{2}\int_{0}^{\pi}\int_{-\pi/2}^{\pi/2} u^2 \sin^2(v) \, dv \, du$. Solving this integral gives $\frac{7\pi^4}{12}$.

Using a Change of Variables, the Double Integral ∬\iint∬ R (x+y)2sin⁡2(x−y)dA( x + y ) ^ { 2 } \sin ^ { 2 } ( x - y ) d A(x+y)2sin2(x−y)dA

Using a Change of Variables, the Double Integral $\iint$ _{R $( x + y ) ^ { 2 } \sin ^ { 2 } ( x - y ) d A$}