Quiz (Change of Variable in Multiple Integrals)

Show all work and justify each step.

(1) Find the Jacobian given and

(2) Let be the region in the -plane that is bounded by the coordinate axes and the line Use the change of variables to compute the integral  

(3) Under the change of variables the quarter circle region in the -plane given by is mapped onto a certain region of the -plane. Evaluate  

(4) Evaluate where is the region bounded by the parallelogram with vertices and

(5) A rotation of the -plane through the fixed angle is given by



Compute the Jacobian Let denote the region bounded by the ellipse Use a rotation of to obtain an integral that is equivalent to



Evaluate the transformed integral.

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