Change of Variables over a Parallelogram

Calculus 3 · Axiom Academy

EXAMPLE Evaluating by Change of Variables Turning a slanted parallelogram into a clean rectangle with the substitution and its Jacobian. Evaluate , where R is the parallelogram bounded by the lines x-y=0 , x-y=2 , x+y=1 , and x+y=3 . Use the change of variables u=x-y , v=x+y . Nicely done — you changed variables to tame both an awkward region and the integrand at once. Match the region: choosing u=x-y , v=x+y turns the four slanted edges into the constant lines that bound the rectangle . Attach the Jacobian: here , the factor that corrects for how the map rescales area. The same recipe — pick u,v to simplify the region, then divide out by |J| — powers every change of variables, including polar, cylindrical, and spherical coordinates.

This is the written version of the interactive lesson above. See the full Calculus 3 course.