Question

Use the given transformation to evaluate the integral.

∬ᵣ (10x+10y)dA, where R is the parallelogram with vertices (-3, 12), (3, -12),(4, -11), and (-2, 13)
x=u+v/5, y = v-4u/5

Answer 1

To evaluate the integral, calculate the Jacobian determinant, express the integral in new variables with the Jacobian, and integrate over the new region corresponding to R in the u-v plane.

Step-by-step explanation:

The student has been asked to evaluate the double integral of the function 10x+10y over a parallelogram region R using a given transformation where x and y are expressed in terms of new variables u and v. The transformation is given by x=u+v/5 and y=v-4u/5. To solve the integral, we need to find the Jacobian determinant of the transformation, which gives us the area scaling factor when changing variables. We then apply the new variables and the Jacobian to the integral, performing the integration over the new region defined by u and v.

Concretely, the steps include:

1. Calculating the Jacobian determinant, J, based on the partial derivatives of x and y concerning u and v.
2. Expressing the integral in terms of u and v, including the Jacobian.
3. Evaluating this new double integral over the region corresponding to the parallelogram R in the u-v plane.

By carrying out these steps, the double integral can be computed over a potentially simpler region, often a rectangle, in the u-v plane.