Question

Evaluate the given double integral by using a stuitable change of coordinates. ∬ (3x−2y) dA,where D is the region bounded by the lines y=−2x+1, y=−2x+3, y=3x/2-4 and y= 3x/2 +2

Answer 1

To evaluate the given double integral using a suitable change of coordinates, we can use a transformation from (x, y) to (u, v) where u = x - y/2 and v = x + y/2. This simplifies the region D and the integrand (3x−2y) dA. The transformed integral can then be evaluated over the new region.

Step-by-step explanation:

To evaluate the given double integral using a suitable change of coordinates, we need to find a transformation that simplifies the given region D and the integrand (3x−2y) dA. In this case, we can use a change of coordinates from (x, y) to (u, v), where:

• u = x - y/2
• v = x + y/2

Using this change of coordinates, the region D transforms into a rectangle in the (u, v) plane, and the integrand simplifies to:

(3/2)(u + v) dudv

We can then evaluate this new double integral over the transformed region.