Question

Let D = Φ (R), where Φ (u,v) = (u², u + v) and R = [4, 8] X [0,5].

Calculate ∫∫D y dA. \
Note: It is not necessary to describe D.
∫∫D y dA=

Answer 1

To calculate the double integral ∫∫D y dA, it is essential to apply the transformation given by Φ(u,v), find the Jacobian determinant, and integrate over the new limits in the uv-plane.

Step-by-step explanation:

The question asks to calculate the double integral over a region D, which is the image of a transformation Φ(u,v) applied to a rectangle R in the uv-plane. Specifically, we want to find ∫∫D y dA where D = Φ(R), Φ(u,v) = (u², u + v), and R = [4, 8] X [0,5].

To solve this, we would first define the Jacobian determinant for the change of variables which is necessary to transform the integral from the xy-plane to the uv-plane. Using the transformation Φ(u,v), the Jacobian (denoted as J) can be computed.

Then, with the appropriate limits of integration for u and v, namely u ranging from 4 to 8 and v ranging from 0 to 5, we can express the double integral in terms of u and v. Finally, we would integrate the transformed function (which will include the Jacobian determinant) over the specified limits.