Q. 40 - Calculus

Question

Evaluate the double integrals in Exercises 39–48. Use suitable transformations as necessary. 

$\int {\int }_{R}xy dA$, where R is the parallelogram with vertices (0, 0), (3, 0), (5, 2), and (2, 2).

Step-by-Step Solution

Answer

$\int {\int }_{R}xy dA=17$

Step 1: Identify the transformation

Use an appropriate change of variables (\(u\)-\(v\) substitution) to simplify the region of integration and the integrand.

Step 2: Compute the Jacobian

The Jacobian \(\frac{\partial(x,y)}{\partial(u,v)}\) accounts for the area scaling in the change of variables: \(dx\,dy = \left|\frac{\partial(x,y)}{\partial(u,v)}\right| du\,dv\).

Step 3: Evaluate

Transform the integral to \(u\)-\(v\) coordinates, simplify, and evaluate.