Change of Variables in Triple Integrals

Proposition (Change of Variables in Triple 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 -space onto under the change of variables and where functions and are continuously differentiable functions on If   then





Example (Change of Variables in Triple Integrals) Compute the volume of an ellipsoid.

    Solution. Assume the ellipsoid is given in standard form by



Let and then the ellipsoid corresponds to the unit sphere Since the Jacobian is
    


and so the volume of an ellipsoid is given by  
    

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