1. MULTIPLE INTEGRALS 16

2. 2 16.8 Triple Integrals in Spherical Coordinates In this section, we will learn how to: Convert rectangular coordinates to spherical ones and use this to evaluate triple integrals. MULTIPLE INTEGRALS

3. 3 SPHERICAL COORDINATE SYSTEM Another useful coordinate system in three dimensions is the spherical coordinate system.  It simplifies the evaluation of triple integrals over regions bounded by spheres or cones.

4. 4 SPHERICAL COORDINATES The spherical coordinates (ρ, θ, Φ) of a point P in space are shown.  ρ = |OP| is the distance from the origin to P.  θ is the same angle as in cylindrical coordinates.  Φ is the angle between the positive z-axis and the line segment OP. Fig. 16.8.1, p. 1041

5. 5 SPHERICAL COORDINATES Note that:  ρ ≥ 0  0 ≤ θ ≤ π Fig. 16.8.1, p. 1041

6. 6 The spherical coordinate system is especially useful in problems where there is symmetry about a point, and the origin is placed at this point. SPHERICAL COORDINATE SYSTEM

7. 7 SPHERE For example, the sphere with center the origin and radius c has the simple equation ρ = c.  This is the reason for the name “spherical” coordinates. Fig. 16.8.2, p. 1042

8. 8 HALF-PLANE The graph of the equation θ = c is a vertical half-plane. Fig. 16.8.3, p. 1042

9. 9 HALF-CONE The equation Φ = c represents a half-cone with the z-axis as its axis. Fig. 16.8.4, p. 1042

10. 10 The relationship between rectangular and spherical coordinates can be seen from this figure. SPHERICAL & RECTANGULAR COORDINATES Fig. 16.8.5, p. 1042

11. 11 From triangles OPQ and OPP’, we have: z = ρ cos Φ r = ρ sin Φ  However, x = r cos θ y = r sin θ SPHERICAL & RECTANGULAR COORDINATES Fig. 16.8.5, p. 1042

12. 12 So, to convert from spherical to rectangular coordinates, we use the equations x = ρ sin Φ cos θ y = ρ sin Φ sin θ z = ρ cos Φ Equations 1 SPH. & RECT. COORDINATES

13. 13 Also, the distance formula shows that: ρ 2 = x 2 + y 2 + z 2  We use this equation in converting from rectangular to spherical coordinates. SPH. & RECT. COORDINATES Equation 2

14. 14 The point (2, π/4, π/3) is given in spherical coordinates. Plot the point and find its rectangular coordinates. SPH. & RECT. COORDINATES Example 1

15. 15 We plot the point as shown. SPH. & RECT. COORDINATES Example 1 Fig. 16.8.6, p. 1042

16. 16 From Equations 1, we have: SPH. & RECT. COORDINATES Example 1

17. 17 Thus, the point (2, π/4, π/3) is in rectangular coordinates. SPH. & RECT. COORDINATES Example 1

18. 18 The point is given in rectangular coordinates. Find spherical coordinates for the point. SPH. & RECT. COORDINATES Example 2

19. 19 From Equation 2, we have: SPH. & RECT. COORDINATES Example 2

20. 20 So, Equations 1 give:  Note that θ ≠ 3π/2 because SPH. & RECT. COORDINATES Example 2

21. 21 Therefore, spherical coordinates of the given point are: (4, π/2, 2π/3) SPH. & RECT. COORDINATES Example 2

22. 22 EVALUATING TRIPLE INTEGRALS WITH SPH. COORDS. In the spherical coordinate system, the counterpart of a rectangular box is a spherical wedge where: a ≥ 0, β – α ≤ 2π, d – c ≤ π

23. 23 Although we defined triple integrals by dividing solids into small boxes, it can be shown that dividing a solid into small spherical wedges always gives the same result. EVALUATING TRIPLE INTEGRALS

24. 24 Thus, we divide E into smaller spherical wedges E ijk by means of equally spaced spheres ρ = ρ i, half-planes θ = θ j, and half- cones Φ = Φ k. EVALUATING TRIPLE INTEGRALS

25. 25 The figure shows that E ijk is approximately a rectangular box with dimensions:  Δρ, ρ i ΔΦ (arc of a circle with radius ρ i, angle ΔΦ)  ρ i sinΦ k Δθ (arc of a circle with radius ρ i sin Φ k, angle Δθ) EVALUATING TRIPLE INTEGRALS Fig. 16.8.7, p. 1043

26. 26 So, an approximation to the volume of E ijk is given by: EVALUATING TRIPLE INTEGRALS

27. 27 In fact, it can be shown, with the aid of the Mean Value Theorem, that the volume of E ijk is given exactly by: where is some point in E ijk. EVALUATING TRIPLE INTEGRALS

28. 28 Let be the rectangular coordinates of that point. EVALUATING TRIPLE INTEGRALS

29. 29 Then, EVALUATING TRIPLE INTEGRALS

30. 30 EVALUATING TRIPLE INTEGRALS However, that sum is a Riemann sum for the function  So, we have arrived at the following formula for triple integration in spherical coordinates.

31. 31 where E is a spherical wedge given by: Formula 3 TRIPLE INTGN. IN SPH. COORDS.

32. 32 Formula 3 says that we convert a triple integral from rectangular coordinates to spherical coordinates by writing: x = ρ sin Φ cos θ y = ρ sin Φ sin θ z = ρ cos Φ TRIPLE INTGN. IN SPH. COORDS.

33. 33 That is done by:  Using the appropriate limits of integration.  Replacing dV by ρ 2 sin Φ dρ dθ dΦ. TRIPLE INTGN. IN SPH. COORDS. Fig. 16.8.8, p. 1044

34. 34 The formula can be extended to include more general spherical regions such as:  The formula is the same as in Formula 3 except that the limits of integration for ρ are g 1 (θ, Φ) and g 2 (θ, Φ). TRIPLE INTGN. IN SPH. COORDS.

35. 35 Usually, spherical coordinates are used in triple integrals when surfaces such as cones and spheres form the boundary of the region of integration. TRIPLE INTGN. IN SPH. COORDS.

36. 36 Evaluate where B is the unit ball: TRIPLE INTGN. IN SPH. COORDS. Example 3

37. 37 As the boundary of B is a sphere, we use spherical coordinates:  In addition, spherical coordinates are appropriate because: x 2 + y 2 + z 2 = ρ 2 Example 3 TRIPLE INTGN. IN SPH. COORDS.

38. 38 So, Formula 3 gives: Example 3 TRIPLE INTGN. IN SPH. COORDS.

39. 39 It would have been extremely awkward to evaluate the integral in Example 3 without spherical coordinates.  In rectangular coordinates, the iterated integral would have been: Note TRIPLE INTGN. IN SPH. COORDS.

40. 40 Use spherical coordinates to find the volume of the solid that lies:  Above the cone  Below the sphere x 2 + y 2 + z 2 = z Example 4 TRIPLE INTGN. IN SPH. COORDS. Fig. 16.8.9, p. 1045

41. 41 Notice that the sphere passes through the origin and has center (0, 0, ½).  We write its equation in spherical coordinates as: ρ 2 = ρ cos Φ or ρ = cos Φ Example 4 TRIPLE INTGN. IN SPH. COORDS. Fig. 16.8.9, p. 1045

42. 42 The equation of the cone can be written as:  This gives: sinΦ = cosΦ or Φ = π/4 Example 4 TRIPLE INTGN. IN SPH. COORDS.

43. 43 Thus, the description of the solid E in spherical coordinates is: Example 4 TRIPLE INTGN. IN SPH. COORDS.

44. 44 The figure shows how E is swept out if we integrate first with respect to ρ, then Φ, and then θ. Example 4 TRIPLE INTGN. IN SPH. COORDS. Fig. 16.8.11, p. 1045

45. 45 The volume of E is: TRIPLE INTGN. IN SPH. COORDS. Example 4

46. 46 This figure gives another look (this time drawn by Maple) at the solid of Example 4. TRIPLE INTGN. IN SPH. COORDS. Fig. 16.8.10, p. 1045