Lecture 20, Monday, 10/13/08: Section 13.3: Double integrals in polar coordinates

Topics

Area element dA in polar coordinates
Computing double integrals for regions defined by polar coordinates
Converting double integrals from rectangular coordinates to polar coordinates
Evaluation of the "Gaussian integral", the integral of exp(-x²) from -infinity to infinity

Read

Section 13.3.

Homework

Section 13.3: 1, 3, 9, 11, 21, 23, 27, 29, 31. Solutions.

Notes

Choosing appropriate coordinates in double integrals. Sometimes you don't have a choice. A problem may specifically ask you to do an integral using rectangular coordinates or polar coordinates (or do it both ways!).
If you do have a choice on which coordinate system to use, here are some general guidelines: First, the shape of the region of integration often makes the choice obvious. If the integral is over a disk or annulus, centered at the origin try to use polar coordinates; for a rectangular or triangular region, use rectangular coordinates. Second, look for some tell-tale signs in the function to be integrated: in particular, a term x² + y² (which is r² in polar) suggests using polar coordinates.
Integration techniques: The most common technique needed for double integrals (especially double integrals in polar coordinates is substitution, and you should be very comfortable in integrating by substitution. (Hint: When doing an integration by substitution, make sure that you also substitute the limits of integration.) Double integrals in polar coordinates often lead to trig integrals. You don't need to know the more esoteric tricks for doing such integrals (like integrating general powers of sin, cos, sec, csc, etc. - if such an integral should come up in a problem, an appropriate hint will be given or the problem may ask you to leave the integral unevaluated), but you should know the basic methods (substitution, using trig identities, etc.). For example, you should be able to handle integrals of (cos x)³ (sin x) (use substitution u = cos x), and integrals of sin² and cos² (use identity relating these to cos(2x)).

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Last modified Wed 15 Oct 2008 07:12:56 AM CDT