14.3-Change-of-variables-polar-coordinates.ppt

Polar coordinates

1. 14.3 Change of Variables,
Polar Coordinates
The equation for this surface is
ρ= sinφ*cos(2θ) (in spherical coordinates)

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

3. R
R

4. 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 rdөthe length is dr
Hence dydx is equivalent to rdrdө

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

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

7. Use the order dr dө
Use the order dөdr

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

9. Example 2 Solution

10. 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

11. 18 solution

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

13. 22 solution

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

15. Problem 24

17. Figure 14.25

18. Figure 14.26

19. Figure 14.27