14.3 Change of Variables,
Polar Coordinates
The equation for this surface is
ρ= sinφ *cos(2θ) (in spherical coordinates)

The region R consists of all points between concentric circles of radii 1 and 3 this is called a Polar sector

R
R

A small rectangle in on the left has an area of dydx
A small piece of area of the portion on the right could be found by using length times width.
The width is rdө the length is dr
Hence dydx is equivalent to rdrdө

In three dimensions, polar
(cylindrical coordinates) look like this.

Change of Variables to Polar Form
Recall: dy dx = r dr dө

Use the order dr dө
Use the order dө dr

Example 2
Let R be the annular region lying between the two circles
Evaluate the integral

Example 2 Solution

Problem 18
Evaluate the integral by converting it to polar coordinates
Note: do this problem in 3 steps
1. Draw a picture of the domain to restate the limits of integration
2. Change the differentials (to match the limits of integration)
3. Use Algebra and substitution to change the integrand

18 solution

Problem 22
Combine the sum of the two iterated integrals into a single iterated integral by converting to polar coordinates. Evaluate the resulting integral.

22 solution

Problem 24
Use polar coordinates to set up and evaluate the double integral

Problem 24

Figure 14.25

Figure 14.26

Figure 14.27