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MULTIPLE INTEGRALS

16. MULTIPLE INTEGRALS. MULTIPLE INTEGRALS. 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. SPHERICAL COORDINATE SYSTEM.

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MULTIPLE INTEGRALS

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  1. 16 MULTIPLE INTEGRALS

  2. MULTIPLE INTEGRALS 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.

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

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

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

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

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

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

  11. SPHERICAL & RECTANGULAR COORDINATES • From triangles OPQ and OPP’, we have: z = ρcos Φr = ρsin Φ • However, x = r cos θy = r sin θ Fig. 16.8.5, p. 1042

  12. SPH. & RECT. COORDINATES Equations 1 • So, to convert from spherical to rectangular coordinates, we use the equations • x = ρsin Φcos θy = ρsin Φsin θz = ρcos Φ

  13. SPH. & RECT. COORDINATES Equation 2 • Also, the distance formula shows that: • ρ2 = x2 + y2 + z2 • We use this equation in converting from rectangular to spherical coordinates.

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

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

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

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

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

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

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

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

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

  23. EVALUATING TRIPLE INTEGRALS • 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.

  24. EVALUATING TRIPLE INTEGRALS • Thus, we divide E into smaller spherical wedges Eijk by means of equally spaced spheres ρ = ρi, half-planes θ = θj, and half-cones Φ = Φk.

  25. EVALUATING TRIPLE INTEGRALS • The figure shows that Eijk is approximately a rectangular box with dimensions: • Δρ, ρi ΔΦ(arc of a circle with radius ρi, angle ΔΦ) • ρi sinΦk Δθ(arc of a circle with radius ρi sin Φk, angle Δθ) Fig. 16.8.7, p. 1043

  26. EVALUATING TRIPLE INTEGRALS • So, an approximation to the volume of Eijkis given by:

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

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

  29. EVALUATING TRIPLE INTEGRALS • Then,

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

  32. TRIPLE INTGN. IN SPH. COORDS. • Formula 3 says that we convert a triple integral from rectangular coordinates to spherical coordinates by writing: x = ρsin Φcos θy = ρsin Φsin θz = ρcos Φ

  33. TRIPLE INTGN. IN SPH. COORDS. • That is done by: • Using the appropriate limits of integration. • Replacing dV by ρ2 sin Φ dρ dθ dΦ. Fig. 16.8.8, p. 1044

  34. TRIPLE INTGN. IN SPH. COORDS. • 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 ρare g1(θ, Φ) and g2(θ, Φ).

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

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

  37. TRIPLE INTGN. IN SPH. COORDS. Example 3 • As the boundary of B is a sphere, we use spherical coordinates: • In addition, spherical coordinates are appropriate because: x2 + y2 + z2 = ρ2

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

  39. TRIPLE INTGN. IN SPH. COORDS. Note • 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:

  40. TRIPLE INTGN. IN SPH. COORDS. Example 4 • Use spherical coordinates to find the volume of the solid that lies: • Above the cone • Below the sphere x2 + y2 + z2 = z Fig. 16.8.9, p. 1045

  41. TRIPLE INTGN. IN SPH. COORDS. Example 4 • Notice that the sphere passes through the origin and has center (0, 0, ½). • We write its equation in spherical coordinates as: ρ2 = ρcos Φor ρ = cos Φ Fig. 16.8.9, p. 1045

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

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

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

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

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

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