1 / 46

MULTIPLE INTEGRALS

15. MULTIPLE INTEGRALS. MULTIPLE INTEGRALS. 15.8 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.

dyanne
Download Presentation

MULTIPLE INTEGRALS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 15 MULTIPLE INTEGRALS

  2. MULTIPLE INTEGRALS 15.8 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.

  5. SPHERICAL COORDINATES • Note that: • ρ≥ 0 • 0 ≤ θ≤ π

  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.

  8. HALF-PLANE • The graph of the equation θ = cis a vertical half-plane.

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

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

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

  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.

  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)

  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.

  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

  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

  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.

  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.

More Related