1. Chapter 15 – Multiple Integrals 15.4 Double Integrals in Polar Coordinates 1 Objectives:  Determine how to express double integrals in polar coordinates Dr. Erickson

2. Polar Coordinates Review Recall from this figure that the polar coordinates (r, θ) of a point are related to the rectangular coordinates (x, y) by the equations ◦ r 2 = x 2 + y 2 ◦ x = r cos θ ◦ y = r sin θ 15.4 Double Integrals in Polar Coordinates2Dr. Erickson

3. Double Integrals in Polar Coordinates Suppose that we want to evaluate a double integral, where R is one of the regions shown here. 15.4 Double Integrals in Polar Coordinates3Dr. Erickson

4. Double Integrals in Polar Coordinates In either case, the description of R in terms of rectangular coordinates is rather complicated but R is easily described by polar coordinates. 15.4 Double Integrals in Polar Coordinates4Dr. Erickson

5. Polar Rectangle The regions in the first figure are special cases of a polar rectangle R = {(r,  ) | a ≤ r ≤ b,  ≤  ≤  } shown here. 15.4 Double Integrals in Polar Coordinates5Dr. Erickson

6. Polar Rectangle To compute the double integral where R is a polar rectangle, we divide: ◦ The interval [a, b] into m subintervals [r i–1, r i ] of equal width ∆r = (b – a)/m. ◦ The interval [ ,  ] into n subintervals [  j–1,  j ] of equal width ∆  = (  –  )/n. 15.4 Double Integrals in Polar Coordinates6Dr. Erickson

7. Polar Subrectangle The “center” of the polar subrectangle R ij = {(r, θ) | r i–1 ≤ r ≤ r i, θ j–1 ≤ θ ≤ θ i } has polar coordinates r i * = ½ (r i–1 + r i ) θ j * = ½ (θ j–1 + θ j ) 15.4 Double Integrals in Polar Coordinates7Dr. Erickson

8. Polar Rectangles The rectangular coordinates of the center of R ij are (r i * cos θ j *, r i * sin θ j *). So, a typical Riemann sum is shown with Equation 1: 15.4 Double Integrals in Polar Coordinates8Dr. Erickson

9. Polar Rectangles If we write g(r, θ) = r f(r cos θ, r sin θ), the Riemann sum in Equation 1 can be written as: ◦ This is a Riemann sum for the double integral 15.4 Double Integrals in Polar Coordinates9Dr. Erickson

10. Polar Rectangles Thus, we have: 15.4 Double Integrals in Polar Coordinates10Dr. Erickson

11. Change to Polar Coordinates 15.4 Double Integrals in Polar Coordinates11Dr. Erickson

12. Change to Polar Coordinates Formula 2 says that we convert from rectangular to polar coordinates in a double integral by: ◦ Writing x = r cos θ and y = r sin θ ◦ Using the appropriate limits of integration for r and θ ◦ Replacing dA by dr dθ Be careful not to forget the additional factor r on the right side of Formula 2. 15.4 Double Integrals in Polar Coordinates12Dr. Erickson

13. Change to Polar Coordinates A classical method for remembering the formula is shown here. ◦ The “infinitesimal” polar rectangle can be thought of as an ordinary rectangle with dimensions r dθ and dr. ◦ So, it has “area” dA = r dr dθ. 15.4 Double Integrals in Polar Coordinates13Dr. Erickson

14. Example 1 Sketch the region whose area is given by the integral and evaluate the integral. 15.4 Double Integrals in Polar Coordinates14Dr. Erickson

15. Example 2 Evaluate the integral by changing to polar coordinates. 15.4 Double Integrals in Polar Coordinates15Dr. Erickson

16. More Complicated Regions What we have done so far can be extended to the more complicated type of region shown here. ◦ It’s similar to the type II rectangular regions considered in Section 15.3 15.4 Double Integrals in Polar Coordinates16Dr. Erickson

17. More Complicated Volumes If f is continuous on a polar region of the form D = {(r, θ) | α ≤ θ ≤ β, h 1 (θ) ≤ r ≤ h 2 (θ)} then 15.4 Double Integrals in Polar Coordinates17Dr. Erickson

18. Volumes and Areas In particular, taking f(x, y) = 1, h 1 (θ) = 0, and h 2 (θ) = h(θ) in the formula, we see that the area of the region D bounded by θ = α, θ = β, and r = h(θ) is: Which agrees with formula 3 in section 10.4. 15.4 Double Integrals in Polar Coordinates18Dr. Erickson

19. Example 3 Use a double integral to find the area of the region. 15.4 Double Integrals in Polar Coordinates19Dr. Erickson

20. Example 4 Use polar coordinates to find the volume of the given solid. 15.4 Double Integrals in Polar Coordinates20Dr. Erickson

21. Example 5 Evaluate the iterated integral by converting to polar coordinates. 15.4 Double Integrals in Polar Coordinates21Dr. Erickson

22. More Examples The video examples below are from section 14.6 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 2 Example 2 ◦ Example 3 Example 3 ◦ Example 4 Example 4 15.4 Double Integrals in Polar Coordinates22Dr. Erickson

23. Demonstrations Feel free to explore these demonstrations below. ◦ Double Integral for Volume Double Integral for Volume 15.4 Double Integrals in Polar Coordinates23Dr. Erickson