MULTIPLE INTEGRALS

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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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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
• 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
• 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 COORDINATES
• Note that:
• ρ≥ 0
• 0 ≤ θ≤ π

Fig. 16.8.1, p. 1041

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

Fig. 16.8.3, p. 1042

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

Fig. 16.8.5, p. 1042

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

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. 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. 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. COORDINATES

Example 1

• We plot the point as shown.

Fig. 16.8.6, p. 1042

SPH. & RECT. COORDINATES

Example 1

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

Example 1

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

Example 2

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

Example 2

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

Example 2

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

Example 2

• Therefore, spherical coordinates of the given point are: (4, π/2 , 2π/3)
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
• 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 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 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 INTEGRALS
• So, an approximation to the volume of Eijkis given by:
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 INTEGRALS
• Let be the rectangular coordinates of that point.
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.

Formula 3

• where E is a spherical wedge given by:
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. 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. 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. 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. COORDS.

Example 3

• Evaluate where B is the unit ball:
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. COORDS.

Example 3

• So, Formula 3 gives:
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. 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. 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. COORDS.

Example 4

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

Example 4

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

Example 4

• The volume of E is:
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