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Chapter 15 – Multiple Integrals15.8 Triple Integrals in Cylindrical Coordinates
15.8 Triple Integrals in Cylindrical Coordinates
Objectives: Use cylindrical coordinates to
solve triple integrals
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Polar Coordinates In plane geometry, the polar coordinate system is used
to give a convenient description of certain curves and regions.
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Polar CoordinatesThe figure enables us
to recall the connection between polar and Cartesian coordinates.
◦ If the point P has Cartesian coordinates (x, y) and polar coordinates (r, θ), then
x = r cos θ y = r sin θ
r2 = x2 + y2 tan θ = y/x
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Cylindrical Coordinates In three dimensions there is a coordinate system, called
cylindrical coordinates, that:
◦ Is similar to polar coordinates.
◦Gives a convenient description of commonly occurring surfaces and solids.
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Cylindrical Coordinates In the cylindrical coordinate system, a point P in three-
dimensional (3-D) space is represented by the ordered triple (r, θ, z), where:
◦ r and θ are polar coordinates of the projection of P onto the xy–plane.
◦ z is the directed
distance from the
xy-plane to P.
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Cylindrical CoordinatesTo convert from cylindrical to rectangular coordinates,
we use the following (Equation 1):
x = r cos θ
y = r sin θ
z = z
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Cylindrical CoordinatesTo convert from rectangular to cylindrical coordinates,
we use the following (Equation 2):
r2 = x2 + y2
tan θ = y/x
z = z
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Example 1Plot the point whose cylindrical coordinates are given.
Then find the rectangular coordinates of the point.
a)
b)
2, ,14
4, ,53
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Example 2 – pg. 1004 # 4Change from rectangular coordinates to cylindrical
coordinates.
a)
b)
2 3,2, 1
4, 3, 2
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Example 3 – pg 1004 # 10Write the equations in cylindrical coordinates.
a)
b)
3 2 6x y z
2 2 2 1x y z
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Cylindrical CoordinatesCylindrical coordinates are useful in problems that
involve symmetry about an axis, and the z-axis is chosen to coincide with this axis of symmetry.
◦ For instance, the axis of the circular cylinder with Cartesian equation x2 + y2 = c2 is the z-axis.
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Cylindrical Coordinates◦ In cylindrical coordinates, this cylinder has
the very simple equation r = c.
◦ This is the reason for the name “cylindrical” coordinates.
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Example 4 – pg 1004 # 12Sketch the solid described by the given inequalities.
02
2r z
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Example 5Sketch the solid whose volume is given by the integral
and evaluate the integral.
2/2 2 9
0 0 0
r
r dz dr d
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Evaluating Triple IntegralsSuppose that E is a type 1 region whose projection D on
the xy-plane is conveniently described in polar coordinates.
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Evaluating Triple Integrals In particular, suppose that f is continuous and
E = {(x, y, z) | (x, y) D, u1(x, y) ≤ z ≤ u2(x, y)}
where D is given in polar coordinates by: D = {(r, θ) | α ≤ θ ≤ β, h1(θ) ≤ r ≤ h2(θ)}
We know from Equation 6 in Section 15.6 that:
2
1
( , )
( , )( , , ) , ,
u x y
u x yE D
f x y z dV f x y z dz dA
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Evaluating Triple IntegralsHowever, we also know how to evaluate double
integrals in polar coordinates.
This is formula 4 for triple integration in cylindrical coordinates.
2 2
1 1
( ) cos , sin
( ) cos , sin
, ,
cos , sin ,
E
h u r r
h u r r
f x y z dV
f r r z r dz dr d
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Evaluating Triple Integrals
It says that we convert a triple integral from rectangular to cylindrical coordinates by:
◦Writing x = r cos θ, y = r sin θ.◦ Leaving z as it is.◦Using the appropriate limits of integration for z, r, and
θ.◦Replacing dV by r dz dr dθ.
2 2
1 1
( ) cos , sin
( ) cos , sin, , cos , sin ,
h u r r
h u r rE
f x y z dV f r r z r dz dr d
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Example 6
2 2 2 2
Evaluate , where is enclosed by the
paraboloid 1 , the cylinder 5,
and the -plane.
z
E
e dV E
z x y x y
xy
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Example 7 – pg. 1004 # 20
2 2 2 2
Evaluate , where is enclosed by the
planes 0 and 5 and by the cylinders
4 and 9.
E
x dV E
z z x y
x y x y
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Example 8 – pg. 1004 # 27Evaluate the integral by changing to
cylindrical coordinates.2
2 2 2
42 2
2 4
y
y x y
xz dz dx dy
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Example 9 – pg. 1004 # 31When studying the formation of mountain ranges,
geologists estimate the amount of work to lift a mountain from sea level. Consider a mountain that is essentially in the shape of a right circular cone. Suppose the weight density of the material in the vicinity of a point P is g(P) and the height is h(P).◦ Find a definite integral that represents the total work
done in forming the mountain.
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Example 9 continuedAssume Mt. Fuji in Japan is the shape of a right circular
cone with radius 62,000 ft, height 12,400 ft, and density a constant 200 lb/ft3. How much work was done in forming Mt. Fuji if the land was initially at sea level?
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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More Examples
The video examples below are from section 15.8 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 3
Dr. Erickson
15.8 Triple Integrals in Cylindrical Coordinates
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Demonstrations
Feel free to explore these demonstrations below.
Exploring Cylindrical CoordinatesIntersection of Two Cylinders
Dr. Erickson
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