1. Chapter 15 – Multiple Integrals
15.9 Triple Integrals in Spherical Coordinates
Objectives:
Use equations to convert rectangular coordinates to spherical coordinates
Use spherical coordinates to evaluate triple integrals
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15.9 Triple Integrals in Spherical Coordinates

2. Spherical Coordinates
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.
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15.9 Triple Integrals in Spherical Coordinates

3. 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.
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15.9 Triple Integrals in Spherical Coordinates

4. Spherical Coordinates
Note:
ρ ≥ 0
0 ≤ θ ≤ π
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15.9 Triple Integrals in Spherical Coordinates

5. 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.
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15.9 Triple Integrals in Spherical Coordinates

6. 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.
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15.9 Triple Integrals in Spherical Coordinates

7. Half-plane The graph of the equation θ = c is a vertical half-plane.
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15.9 Triple Integrals in Spherical Coordinates

8. Half-cone
The equation Φ = c represents a half-cone with the z-axis as its axis.
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15.9 Triple Integrals in Spherical Coordinates

9. Spherical and Rectangular Coordinates
The relationship between rectangular and spherical coordinates can be seen from this figure.
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
The distance formula shows that:
ρ2 = x2 + y2 + z2
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15.9 Triple Integrals in Spherical Coordinates

10. Spherical and Rectangular Coordinates
To convert from rectangular to
spherical coordinates,
we use the equations
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15.9 Triple Integrals in Spherical Coordinates

11. Example 1
Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point. a) b)
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15.9 Triple Integrals in Spherical Coordinates

12. Example 2 Change from rectangular to spherical coordinates. a) b)
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15.9 Triple Integrals in Spherical Coordinates

13. Example 3 Write the equation in spherical coordinates. a) b)
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15.9 Triple Integrals in Spherical Coordinates

14. Evaluating Triple Integrals
In the spherical coordinate system, the counterpart of a rectangular box is a spherical wedge
where:
a ≥ 0, β – α ≤ 2π, d – c ≤ π
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15.9 Triple Integrals in Spherical Coordinates

15. Visualization A region in spherical coordinates
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15.9 Triple Integrals in Spherical Coordinates

16. 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.
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15.9 Triple Integrals in Spherical Coordinates

17. 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)
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15.9 Triple Integrals in Spherical Coordinates

18. Evaluating Triple Integrals
Using the idea of Riemann Sum, we can write the sum as
where and
is some point in Eijk.
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15.9 Triple Integrals in Spherical Coordinates

19. Evaluating Triple Integrals
Which leads to the following integral called formula 3:
where E is a spherical wedge given by:
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15.9 Triple Integrals in Spherical Coordinates

20. Spherical Coordinates
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
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15.9 Triple Integrals in Spherical Coordinates

21. Spherical Coordinates
That is done by:
Using the appropriate limits of integration.
Replacing dV by ρ2 sin Φ dρ dθ dΦ.
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15.9 Triple Integrals in Spherical Coordinates

22. Triple Integrals in Spherical Coordinates
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 g1(θ, Φ) and g2(θ, Φ).
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15.9 Triple Integrals in Spherical Coordinates

23. Triple Integrals in Spherical Coordinates
Usually, spherical coordinates are used in triple integrals when surfaces such as cones and spheres form the boundary of the region of integration.
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15.9 Triple Integrals in Spherical Coordinates

24. Example 4
Sketch the solid whose volume is given by the integral and evaluate the integral.
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15.9 Triple Integrals in Spherical Coordinates

25. Example 5
Set up the triple integral of an arbitrary continuous function f (x, y, z) in cylindrical or spherical coordinates over the solid shown.
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15.9 Triple Integrals in Spherical Coordinates

26. Example 6 Use spherical coordinates.
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15.9 Triple Integrals in Spherical Coordinates

27. Example 7 Use spherical coordinates.
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15.9 Triple Integrals in Spherical Coordinates

28. Example 8
Use cylindrical or spherical coordinates, whichever seems more appropriate.
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15.9 Triple Integrals in Spherical Coordinates

29. Example 9 Evaluate the integral by changing to spherical coordinates.
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15.9 Triple Integrals in Spherical Coordinates

30. More Examples
The video examples below are from section in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length.
Example 3
Example 4
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15.9 Triple Integrals in Spherical Coordinates

31. Demonstrations Feel free to explore these demonstrations below.
Spherical Coordinates
Exploring Spherical Coordinates
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15.9 Triple Integrals in Spherical Coordinates