Question

Given the integral ∬ Rex+y dA where the region R is bounded by the graphs y=−x+4,y=−x+2,x=0,y=0 a) Graph the region R b) Use the transformation u=x+y,v=x−y to graph the region R' c) Solve the equations for x and y in terms of u and v d) Calculate the Jacobean e) Rewrite the integral and calculate

Answer 1

The solution involves graphing the original region R, transforming it using given equations, solving for x and y in terms of the new variables, calculating the Jacobian, and rewriting and evaluating the double integral in the new variables.

Step-by-step explanation:

To address the question on how to evaluate the double integral over region R given by the integral ∫∫ Rex+y dA, where R is bounded by the graphs y = -x + 4, y = -x + 2, x = 0, and y = 0, we will follow a step-by-step approach:

• Graph the region R: The region is a trapezoidal area confined between the lines described.
• Transformation u = x + y, v = x - y: We will use this transformation to graph the transformed region R' where the bounds become straight lines along the u and v axes.
• Solve for x and y: To express x and y in terms of u and v, we add and subtract the two given transformation equations to get x = (u + v)/2 and y = (u - v)/2.
• Calculate the Jacobian: The Jacobian of the transformation (∂(x,y)/∂(u,v)) is found to be the determinant of the matrix of partial derivatives, which equals 1/2 in this case.
• Rewrite the integral: Express the double integral in terms of the new variables u and v, taking into account the Jacobian, and calculate the integral over the transformed region R'.