MULTIPLE INTEGRALS

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MULTIPLE INTEGRALS
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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.
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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.

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SPHERICAL COORDINATES

• The spherical coordinates (?, ?, F) of a point P
  in space are shown.
• ? OP is the distance from the origin to P.
• ? is the same angle as in cylindrical
  coordinates.
• F is the angle between the positive z-axis and
  the line segment OP.

Fig. 16.8.1, p. 1041
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SPHERICAL COORDINATES

• Note that
• ? 0
• 0 ? p

Fig. 16.8.1, p. 1041
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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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SPHERE

• For example, the sphere with center the origin
  and radius c has the simple equation ? c.
• This is the reason for the name spherical
  coordinates.

Fig. 16.8.2, p. 1042
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HALF-PLANE

• The graph of the equation ? c is a vertical
  half-plane.

Fig. 16.8.3, p. 1042
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HALF-CONE

• The equation F c represents a half-cone with
  the z-axis as its axis.

Fig. 16.8.4, p. 1042
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SPHERICAL RECTANGULAR COORDINATES

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

Fig. 16.8.5, p. 1042
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SPHERICAL RECTANGULAR COORDINATES

• From triangles OPQ and OPP, we have z ?
  cos F r ? sin F
• However, x r cos ? y r sin ?

Fig. 16.8.5, p. 1042
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SPH. RECT. COORDINATES
Equations 1

• So, to convert from spherical to rectangular
  coordinates, we use the equations
• x ? sin F cos ? y ? sin F sin ? z ? cos F

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SPH. RECT. COORDINATES
Equation 2

• Also, the distance formula shows that
• ?2 x2 y2 z2
• We use this equation in converting from
  rectangular to spherical coordinates.

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SPH. RECT. COORDINATES
Example 1

• The point (2, p/4, p/3) is given in spherical
  coordinates.
• Plot the point and find its rectangular
  coordinates.

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SPH. RECT. COORDINATES
Example 1

• We plot the point as shown.

Fig. 16.8.6, p. 1042
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SPH. RECT. COORDINATES
Example 1

• From Equations 1, we have

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SPH. RECT. COORDINATES
Example 1

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

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SPH. RECT. COORDINATES
Example 2

• The point is given in
  rectangular coordinates.
• Find spherical coordinates for the point.

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SPH. RECT. COORDINATES
Example 2

• From Equation 2, we have

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SPH. RECT. COORDINATES
Example 2

• So, Equations 1 give
• Note that ? ? 3p/2 because

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SPH. RECT. COORDINATES
Example 2

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

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EVALUATING TRIPLE INTEGRALS WITH SPH. COORDS.

• In the spherical coordinate system, the
  counterpart of a rectangular box is a spherical
  wedgewhere
• a 0, ß a 2p, d c p

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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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EVALUATING TRIPLE INTEGRALS

• Thus, we divide E into smaller spherical wedges
  Eijk by means of equally spaced spheres ? ?i,
  half-planes ? ?j, and half-cones F Fk.

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EVALUATING TRIPLE INTEGRALS

• The figure shows that Eijk is approximately a
  rectangular box with dimensions
• ??, ?i ?F (arc of a circle with radius ?i,
  angle ?F)
• ?i sinFk ?? (arc of a circle with radius ?i sin
  Fk, angle ??)

Fig. 16.8.7, p. 1043
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EVALUATING TRIPLE INTEGRALS

• So, an approximation to the volume of Eijk is
  given by

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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.

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EVALUATING TRIPLE INTEGRALS

• Let be the rectangular
  coordinates of that point.

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EVALUATING TRIPLE INTEGRALS

• Then,

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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.

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TRIPLE INTGN. IN SPH. COORDS.
Formula 3

• where E is a spherical wedge given by

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TRIPLE INTGN. IN SPH. COORDS.

• Formula 3 says that we convert a triple integral
  from rectangular coordinates to spherical
  coordinates by writing x ? sin F cos ?
  y ? sin F sin ? z ? cos F

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TRIPLE INTGN. IN SPH. COORDS.

• That is done by
• Using the appropriate limits of integration.
• Replacing dV by ?2 sin F d? d? dF.

Fig. 16.8.8, p. 1044
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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 ? are g1(?,
  F) and g2(?, F).

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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.

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TRIPLE INTGN. IN SPH. COORDS.
Example 3

• Evaluate where B is
  the unit ball

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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 ?2

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TRIPLE INTGN. IN SPH. COORDS.
Example 3

• So, Formula 3 gives

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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

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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
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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 ?2 ? cos F or ?
  cos F

Fig. 16.8.9, p. 1045
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TRIPLE INTGN. IN SPH. COORDS.
Example 4

• The equation of the cone can be written as
• This gives sinF cosF or F p/4

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TRIPLE INTGN. IN SPH. COORDS.
Example 4

• Thus, the description of the solid E in
  spherical coordinates is

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TRIPLE INTGN. IN SPH. COORDS.
Example 4

• The figure shows how E is swept out if we
  integrate first with respect to ?, then F, and
  then ?.

Fig. 16.8.11, p. 1045
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TRIPLE INTGN. IN SPH. COORDS.
Example 4

• The volume of E is

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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