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 Draw a picture of the domain to restate the limits of integration Change the differentials (to match the limits of integration) 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