Question

Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral over the region \(R\). \(\int_{R} \int x y d A\) \(R:\) rectangle with vertices (0,0),(0,5),(3,5),(3,0)

Short answer

The result of the integral over the region R is 56.25

Step-by-step solution

Initial double integral setup

The given rectangular region \( R \) has vertices (0,0),(0,5),(3,5),(3,0). The double integral over this region is: \(\int_{0}^{3} \int_{0}^{5} x y \, dy \, dx \)

Swap the order of integration

The other order of integration will be integrating with respect to x first: \(\int_{0}^{5} \int_{0}^{3} x y \, dx \, dy \)

Evaluate the easier double integral

The double integral that integrates with respect to y first is easier to evaluate, as the \(\int x y \, dx \) part of the function can be evaluated directly. We find this as: \(\int_{0}^{5} \left[ \frac{1}{2} x^{2} y \right]_{x=0}^{x=3} dy = \int_{0}^{5} \frac{9}{2} y dy = \left[ \frac{9}{4} y^{2} \right]_{y=0}^{y=5} = \frac{225}{4} = 56.25 \)