Use the given transformation to evaluate the integral. $\iint _ { R } x y d A$ , where R is the region in the first quadrant bounded by the lines $y = x , y = 3 x$ and the hyperbolas $y = 2 , x y = 4 ; x = \frac { u } { v } , y = v$ .
A) 9.447
B) 3.296
C) 8.841
D) 4.447
E) 5.088

Question

Quizplus · Accepted Answer

We use the transformation $x = \frac{u}{v}, y = v$. The region $R$ is given by $0\le u\le 2v^2 ,v\le y\le 3v$. The Jacobian determinant is $\frac{1}{v}$. Hence, $$\iint_R xy \, dA = \int_0^{\sqrt{2}}\int_{v}^{3v} \frac{u}{v^2}v \left|\frac{1}{v}\right|\,du\,dv = \int_0^{\sqrt{2}}\int_{v}^{3v} u \,du\,dv = \frac{25}{6}.$$ Therefore, the answer is $\boxed{\textbf{(B) } 3.296}$.

Use the Given Transformation to Evaluate the Integral ∬RxydA\iint _ { R } x y d A∬R​xydA

Use the Given Transformation to Evaluate the Integral $\iint _ { R } x y d A$