evaluating multiple integrals: change of variables
(1.3 hours to learn)
Summary
A useful trick for computing multiple integrals is to find a simpler parameterization of the region and apply the change of variables formula.
Context
This concept has the prerequisites:
- determinant and volume
- evaluating multiple integrals: polar coordinates (Polar coordinates are an instructive example.)
- determinant (The change of variables formula involves the determinant of the Jacobian.)
- linear approximation (The change of variables formula involves a linear approximation to the transformation.)
Core resources (read/watch one of the following)
-Free-
→ MIT Open Courseware: Multivariable Caclulus (2010)
Video lectures for MIT's introductory multivariable calculus class.
Location:
Session 53, "Change of variables"
-Paid-
→ Multivariable Calculus
An introductory multivariable calculus textbook.
Location:
Section 14.9, "Change of variables in multiple integrals," pages 1001-1007
→ Multivariable Mathematics
A textbook on linear algebra and multivariable calculus with proofs.
Location:
Section 7.6, "Change of Variables Theorem," pages 324-331
Other notes:
- You can skip the proofs as far as this node is concerned.
See also
- Some particular examples of this trick: We use the same change of variables formula to determine the PDF of a function of a continuous random variable .