Question

Consider the following integral and transformation. (3x 9y) da r , where r is the parallelogram with vertices (-3, 6), (3, -6), (5, -4), and (-1, 8); x = 1 3 (u v), y = 1 3 (v - 2u) use the transformation to write an equivalent integral in terms of u and v. 9 -9 a 0 incorrect: your answer is incorrect. dv du a = evaluate the integral.

Answer 1

To rewrite the integral in terms of u and v, calculate the Jacobian determinant from the given transformations, substitute it into the integral as the new area element, and then evaluate the new integral within the bounds defined by the transformed parallelogram.

Step-by-step explanation:

The student is asked to use a given transformation to rewrite a double integral over a parallelogram in terms of new variables u and v. Given transformations are x = 1/3(u+v) and y = 1/3(v - 2u). The area element da in the new coordinates will be the determinant of the Jacobian matrix of this transformation. To find this determinant, we calculate the partial derivatives of x and y with respect to u and v:

Jacobian matrix:
| ∂x/∂u ∂x/∂v |
| ∂y/∂u ∂y/∂v |

Determinant of Jacobian matrix = (∂x/∂u)(∂y/∂v) - (∂x/∂v)(∂y/∂u)

Substitute the determined Jacobian into the integral, and evaluate the new integral in terms of u and v. The new limits of integration will correspond to the vertices of the parallelogram in the u-v plane.