Question

Use the given transformation to evaluate the integral. (12x + 8y) dA R , where R is the parallelogram with vertices (−1, 3), (1, −3), (2, −2), and (0, 4) ; x = 1/ 4 (u + v), y = 1/ 4 (v − 3u)

Answer 1

To evaluate the integral, we substitute the given transformations for x and y into the integrand, and then change the limits of integration to correspond to the new variables.

Step-by-step explanation:

To evaluate the integral using the given transformation, we can substitute the values of x and y in terms of u and v into the integrand, and then change the limits of integration to correspond to the new variables.

Using the given transformations x = 1/4(u + v) and y = 1/4(v - 3u), we have (12x + 8y) dA = (12(1/4(u + v)) + 8(1/4(v - 3u))) d(uv).

To find the new limits of integration, we substitute the original x and y values for the vertices of the parallelogram into the given transformations. The new limits are u = -1, v = -3 to u = 2, v = -2.