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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

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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.

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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

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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.

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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.

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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

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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

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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

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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

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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

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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.

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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.

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Example 4 – pg 1004 # 12Sketch the solid described by the given inequalities.

02

2r z

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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

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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.

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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

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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

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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

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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

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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

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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

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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.

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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?

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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

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Demonstrations

Feel free to explore these demonstrations below.

Exploring Cylindrical CoordinatesIntersection of Two Cylinders

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