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 dydxA 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 steps1. Draw a picture of the domain to restate

the limits of integration2. 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