Calc 3 Lecture NotesSection 13.3Page 1 of 6

Section 13.3: Double Integrals in Polar Coordinates

Big idea: Making a change of variables to polar coordinates (x = rcos(), y = rsin()) can make many integrals easier to compute. The differential area element in polar coordinates is
dA = rdrd.

Big skill: You should be able to recognize integrals that would be simpler to evaluate in polar coordinates and then evaluate them using polar coordinates.

Sometimes the double integral is difficult to evaluate in rectangular coordinates because:

  • It is hard to find the antiderivative off(x, y).
  • The limits of integration in x and y are nasty.

Many times, converting to a different coordinate system eliminates these difficulties. This section focuses on using the polar coordinate system to simplify double integrals.

Cartesian polar conversion formulas:

x = rcos() and y = rsin(), which implies x2 + y2 = r2

In addition to making the substitution f(x, y) = f(rcos(), rsin()),we also need to:

  • Describe the region of integration R in terms of r and , which is usually pretty easy to do.
  • Write the differential element of surface area dA in terms of r and , which is just a formula derived below that you have to remember.

Differential Area Element in Polar Coordinates

In rectangular coordinates, the differential area element comes from thinking about subdividing the region of integration into little rectangles. Thus, A = xydA = dxdy.

In polar coordinates, the differential area element comes from thinking about subdividing the region of integration into little “polar regions.”

Thus, the area of any given polar region is given by:

Thus, ArrdA = rdrd.

This leads us to:

Theorem 3.1: Fubini’s Theorem

If f(r, ) is continuous on the region R = {(r, ) |  ≤  ≤ , g1() ≤ r ≤ g2()}, where
0 ≤ g1() ≤ g2() for all   [, ], then:

.

Practice:

  1. Compute the volume of the solid bounded by the surface and the three coordinate planes. Compare your answer and ease of calculation with this same problem from 13.2.
  1. Compute the area of the cardioid r = 2 + 2cos().
  1. Compute the volume of the hyperboloid of one sheet defined by the equation
    x2 + y2 – z2 = 1 for -2 ≤ z ≤ 2.
  1. It is very important in statistics to know how to normalize the normal distribution function . Find the normalization constant to make this a valid probability distribution function (pdf). If there is time, repeat for the “full-blown” normal pdf.

Theorem Not-Important-Enough-To-Get-A-Number-Because-No-One-Ever-Uses-It:

If f(r, ) is continuous on the region R = {(r, ) | h1(r)≤  ≤ h2(r)}, a ≤ r ≤ b}, where
h1(r) ≤ h2(r) for all r  [a, b], then:

.