Final answer:
The question involves using a linear transformation to evaluate a double integral over a parallelogram. The process includes finding the new integration region, computing the Jacobian, and then performing the actual integration.
Step-by-step explanation:
The student is tasked with evaluating the integral ∫∫R(15x+10y)dA, where R is a parallelogram defined by its vertices. To perform this integral, it is necessary to use the linear transformation given by x=1/5(u+v), and y=15(v−4u). This substitution transforms the original parallelogram into a new region, likely to be more manageable for integration, such as a rectangle in the uv-plane.
Firstly, one should find the corresponding region in the uv-plane that maps to the parallelogram R in the xy-plane using the transformation equations. Then, set up the double integral over the new uv-region. It's important to also compute the Jacobian determinant of the transformation, which is the absolute value of the determinant of the partial derivatives of x and y with respect to u and v. The Jacobian is used to scale the area element dA appropriately after the transformation.
Once the integral is set up with the correct limits and integrand, which now includes the Jacobian, the next step is to perform the actual integration. This may involve standard techniques such as evaluating the integral over u first and then over v, or vice versa, depending on the specifics of the new region's limits.