Chapter 15 – Multiple Integrals 15.9 Triple Integrals in Spherical Coordinates 1 Objectives: Use...
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Chapter 15 – Multiple Integrals 15.9 Triple Integrals in Spherical Coordinates 15.9 Triple Integrals in Spherical Coordinates 1 Objectives: Use equations to convert rectangular coordinates to spherical coordinates Use spherical coordinates to evaluate triple integrals Dr. Erickson
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Transcript of Chapter 15 – Multiple Integrals 15.9 Triple Integrals in Spherical Coordinates 1 Objectives: Use...
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
2
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.
Dr. Erickson
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.Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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Spherical CoordinatesNote:
◦ ρ ≥ 0
◦ 0 ≤ θ ≤ π
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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.
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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.
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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Half-planeThe graph of the equation θ = c is a vertical half-plane.
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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Half-coneThe equation Φ = c represents a half-cone with the
z-axis as its axis.
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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Example 2Change from rectangular to spherical coordinates.
a)
b)
0, 3,1
1,1, 6
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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VisualizationA region in spherical coordinates
Dr. Erickson
15.9 Triple Integrals 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.
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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)
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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Spherical CoordinatesThat is done by:
◦Using the appropriate limits of integration.
◦Replacing dV by ρ2 sin Φ dρ dθ dΦ.
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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.
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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.
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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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
Dr. Erickson
15.9 Triple Integrals in Spherical Coordinates
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Demonstrations
Feel free to explore these demonstrations below.
Spherical CoordinatesExploring Spherical Coordinates
Dr. Erickson
