Use the given transformation to evaluate the integral.
-$$\begin{array} { l }
\mathrm { u } = - 6 \mathrm { x } + \mathrm { y } , \mathrm { v } = 4 \mathrm { x } + \mathrm { y } ; \
\quad \int _ { \mathrm { R } } \int ( \mathrm { y } - 6 \mathrm { x } ) ( 4 \mathrm { x } + \mathrm { y } ) \mathrm { dxdy } ,
\end{array}$$
where $R$ is the parallelogram bounded by the lines $y = 6 x + 5 , y = 6 x + 9 , y = - 4 x + 3 , y = - 4 x + 10$
A) $\frac { 637 } { 5 }$
B) $\frac { 1274 } { 5 }$
C) 25,480
D) 12,740

Question

Use the given transformation to evaluate the integral.
-$$\begin{array} { l } 
\mathrm { u } = - 6 \mathrm { x } + \mathrm { y } , \mathrm { v } = 4 \mathrm { x } + \mathrm { y } ; \
\quad \int _ { \mathrm { R } } \int ( \mathrm { y } - 6 \mathrm { x } ) ( 4 \mathrm { x } + \mathrm { y } ) \mathrm { dxdy } ,
\end{array}$$
where $R$ is the parallelogram bounded by the lines $y = 6 x + 5 , y = 6 x + 9 , y = - 4 x + 3 , y = - 4 x + 10$
A) $\frac { 637 } { 5 }$
B) $\frac { 1274 } { 5 }$
C) 25,480
D) 12,740

Quizplus · Accepted Answer

The given transformation simplifies the integral by directly substituting $u = y - 6x$ and $v = 4x + y$, turning the integral into a simpler form in terms of $u$ and $v$. The bounds of integration become the transformed coordinates of the parallelogram R in the $uv$-plane, which simplifies the calculation. The correct answer involves calculating the integral in the $uv$-plane, taking into account the Jacobian of the transformation, which accounts for the change in area element from $dxdy$ to $dudv$. The calculation leads to the result $\frac{637}{5}$.

Use the Given Transformation to Evaluate the Integral Where RRR Is the Parallelogram Bounded by the Lines

Use the Given Transformation to Evaluate the Integral Where $R$ Is the Parallelogram Bounded by the Lines