evaluating multiple integrals: change of variables
(1.3 hours to learn)
A useful trick for computing multiple integrals is to find a simpler parameterization of the region and apply the change of variables formula.
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)
→ MIT Open Courseware: Multivariable Caclulus (2010)
Video lectures for MIT's introductory multivariable calculus class.
Location: Session 53, "Change of variables"
→ 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
- You can skip the proofs as far as this node is concerned.
- Some particular examples of this trick:PDF of a function of a continuous random variable .
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