Post on 22-Dec-2015
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Chapter 15 – Multiple Integrals15.9 Triple Integrals in Spherical Coordinates
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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Spherical CoordinatesAnother 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 (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.Dr. Erickson
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Spherical CoordinatesNote:
◦ ρ ≥ 0
◦ 0 ≤ θ ≤ π
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Spherical Coordinate SystemThe 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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SphereFor 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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Half-planeThe graph of the equation θ = c is a vertical half-plane.
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Half-coneThe equation Φ = c represents a half-cone with the
z-axis as its axis.
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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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Spherical and Rectangular Coordinates
To convert from rectangular to
spherical coordinates,
we use the equations
2 2 2 2
cossin
cos
x y z
x
z
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Example 1Plot the point whose spherical coordinates are given.
Then find the rectangular coordinates of the point.
a)
b)
5, ,2
34, ,
4 3
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Example 2Change from rectangular to spherical coordinates.
a)
b)
0, 3,1
1,1, 6
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Example 3Write the equation in spherical coordinates.
a)
b)
2 2 22 0x x y z
2 3 1x y z
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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
, , , ,E a b c d
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VisualizationA region in spherical coordinates
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Evaluating Triple IntegralsAlthough 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 IntegralsThe 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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Evaluating Triple IntegralsUsing the idea of Riemann Sum, we can write the sum
as
where and
is some point in Eijk.
* * *
, ,1 1 1
, , lim , ,l m n
ijk ijk ijk ijkl m n
i j kE
f x y z dV f x y z V
, ,i i i 2sini kijkV
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Evaluating Triple IntegralsWhich leads to the following integral called formula 3:
where E is a spherical wedge given by:
2
, ,
sin cos , sin sin , cos sin
E
d b
c a
f x y z dV
f d d d
, , , ,E a b c d
Dr. Erickson
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Spherical CoordinatesFormula 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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Spherical CoordinatesThat is done by:
◦Using the appropriate limits of integration.
◦Replacing dV by ρ2 sin Φ dρ dθ dΦ.
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Triple Integrals in Spherical CoordinatesThe 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).
1 2, , , , , ,E c d g g
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Triple Integrals in Spherical CoordinatesUsually, 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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Example 4Sketch the solid whose volume is given by the integral
and evaluate the integral.
/6 /2 32
0 0 0
sin d d d
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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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Example 6Use spherical coordinates.
2 2
2 2 2
Evaluate 9 , where is the
solid hemisphere 9, 0.
H
x y dV H
x y z z
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Example 7Use spherical coordinates.
2 2 2 2 2 2
Evaluate , where lies between the
spheres 1 and 4
in the first octant.
E
z dV E
x y z x y z
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Example 8Use cylindrical or spherical coordinates,
whichever seems more appropriate.
2 2
2 2 2
Find the volume and centroid of the solid
that lies above the cone and
below the sphere 1.
E z x y
x y z
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Example 9Evaluate the integral by changing to
spherical coordinates.
2 22
2 2 2
2 42 4 32 2 2 2
2 4 2 4
x yx
x x y
x y z dzdydx
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More Examples
The video examples below are from section 15.9 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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Demonstrations
Feel free to explore these demonstrations below.
Spherical CoordinatesExploring Spherical Coordinates
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