Change of Variables in Double Integrals

Proposition (Change of Variables in Double Integrals) Let be a continuous function on a region in the and let be a one-to-one transformation that maps the region in the -plane onto under the change of variables where and are continuously differentiable functions on If   then

Example (Area of an Ellipse) Use a change of variables to compute the area of an ellipse.

    Solution. Assume the ellipse is given in standard form by Let and then the ellipse in the -plane corresponds to the unit circle in the -plane. Since and the Jacobian is and so the area of an ellipse is given by  
    

Example (Change of Variables in Double Integrals) Use a change of variables to evaluate the integral



where is the trapezoidal region with vertices and
    
    Solution. Since it is not easy to integrate we make a change of variables suggested by the form of namely: and   Then, since


        
the Jacobian is,



To find the region in the -plane corresponding to we note that the sides of lie on the lines and using the rules for the transformation, and the images of the lines in the are



Thus the region is the trapezoidal region with vertices and that is



The transformation from to is,




Therefore,

Example (Change of Variables in Double Integrals) Use a change of variables to evaluate the integral



where is the region bounded by the lines and
    
    Solution. Let and Then solving for and produces

and

We compute the Jacobian of and



The bounds under the transformation are



Let's consider the region in the -plane as vertically simple as follows,

Example (Change of Variables in Double Integrals) Use a change of variables to evaluate the integral



where is the region bounded by the square with vertices and
    
    Solution. The region is bounded by the lines and Let and then solving for and produces

and
    
    We compute the Jacobian of and



The bounds under the transformation are



Let's consider the region in the -plane as horizontally simple as follows,

Example (Change of Variables in Double Integrals) Use a change of variables to evaluate the integral



where is the region bounded by the parallelogram with vertices and
    
    Solution. The boundary lines of the parallelgram are and Let and with boundary lines and Solving for and produces

and
    
Since the Jacobian is
    


we have

    In the next example we will make use of



because it is easier not to solve for and in terms of and

Example (Change of Variables in Double Integrals) Use a change of variables to evaluate the integral



where is the region bounded by the hyperbolas and
    
    Solution. Let and Then



Since, we have



Thus,



and so



Now then,

Example (Change of Variables in Double Integrals) Use a change of variables to evaluate the integral



where is the triangular region bounded by the vertices
    
    Solution. Let and so that and Then the Jacobian is
    


The given the region is bounded by the lines and which transform into and Therefore,

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Cite this as:
Change Of Variables In Double Integrals
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/change-of-variables-in-double-integrals.html