Question

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

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

Short answer

$\int {\int }_{R}xydA=17$

Step-by-step solution

Draw the region

Plot the given points to form the region and name the vertices.

In the region R, the equations of boundary curves are,

$$\begin{gathered} AB:y=0 \\ BC:x-y=3 \\ CD:y=2 \\ DA:x-y=0 \end{gathered}$$

Consider the new set of variables defined as,

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Add the above equations,