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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

16

MULTIPLE INTEGRALS


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
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.


Spherical coordinates
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


Spherical coordinates1
SPHERICAL COORDINATES

  • Note that:

    • ρ≥ 0

    • 0 ≤ θ≤ π

Fig. 16.8.1, p. 1041


Spherical coordinate system1
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.


Sphere
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


Half plane
HALF-PLANE

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

Fig. 16.8.3, p. 1042


Half cone
HALF-CONE

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

Fig. 16.8.4, p. 1042


Spherical rectangular coordinates
SPHERICAL & RECTANGULAR COORDINATES

  • The relationship between rectangular and spherical coordinates can be seen from this figure.

Fig. 16.8.5, p. 1042


Spherical rectangular coordinates1
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


Sph rect coordinates
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


Sph rect coordinates1
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.


Sph rect coordinates2
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.


Sph rect coordinates3
SPH. & RECT. COORDINATES

Example 1

  • We plot the point as shown.

Fig. 16.8.6, p. 1042


Sph rect coordinates4
SPH. & RECT. COORDINATES

Example 1

  • From Equations 1, we have:


Sph rect coordinates5
SPH. & RECT. COORDINATES

Example 1

  • Thus, the point (2, π/4, π/3) is in rectangular coordinates.


Sph rect coordinates6
SPH. & RECT. COORDINATES

Example 2

  • The point is given in rectangular coordinates.

  • Find spherical coordinates for the point.


Sph rect coordinates7
SPH. & RECT. COORDINATES

Example 2

  • From Equation 2, we have:


Sph rect coordinates8
SPH. & RECT. COORDINATES

Example 2

  • So, Equations 1 give:

    • Note that θ≠ 3π/2 because


Sph rect coordinates9
SPH. & RECT. COORDINATES

Example 2

  • Therefore, spherical coordinates of the given point are: (4, π/2 , 2π/3)


Evaluating triple integrals with sph coords
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


Evaluating triple integrals
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.


Evaluating triple integrals1
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.


Evaluating triple integrals2
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


Evaluating triple integrals3
EVALUATING TRIPLE INTEGRALS

  • So, an approximation to the volume of Eijkis given by:


Evaluating triple integrals4
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.


Evaluating triple integrals5
EVALUATING TRIPLE INTEGRALS

  • Let be the rectangular coordinates of that point.



Evaluating triple integrals7
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.


Triple intgn in sph coords
TRIPLE INTGN. IN SPH. COORDS.

Formula 3

  • where E is a spherical wedge given by:


Triple intgn in sph coords1
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


Triple intgn in sph coords2
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


Triple intgn in sph coords3
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).


Triple intgn in sph coords4
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.


Triple intgn in sph coords5
TRIPLE INTGN. IN SPH. COORDS.

Example 3

  • Evaluate where B is the unit ball:


Triple intgn in sph coords6
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


Triple intgn in sph coords7
TRIPLE INTGN. IN SPH. COORDS.

Example 3

  • So, Formula 3 gives:


Triple intgn in sph coords8
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:


Triple intgn in sph coords9
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


Triple intgn in sph coords10
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


Triple intgn in sph coords11
TRIPLE INTGN. IN SPH. COORDS.

Example 4

  • The equation of the cone can be written as:

    • This gives: sinΦ = cosΦor Φ = π/4


Triple intgn in sph coords12
TRIPLE INTGN. IN SPH. COORDS.

Example 4

  • Thus, the description of the solid Ein spherical coordinates is:


Triple intgn in sph coords13
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


Triple intgn in sph coords14
TRIPLE INTGN. IN SPH. COORDS.

Example 4

  • The volume of E is:


Triple intgn in sph coords15
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