Transcript Slide 1

Chapter 15 – Multiple Integrals
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 Coordinates

The 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
tan θ = y/x
r2 = x2 + y 2
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 threedimensional (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 Coordinates

To 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 Coordinates

To 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
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Example 1

Plot the point whose cylindrical coordinates are given.
Then find the rectangular coordinates of the point.

a)

b) 
  
 2, ,1
 4 


4,

,5


3


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15.8 Triple Integrals in Cylindrical
Coordinates
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Example 2 – pg. 1004 # 4

Change from rectangular coordinates to cylindrical
coordinates.



a) 2 3, 2, 1

b)
 4, 3, 2
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15.8 Triple Integrals in Cylindrical
Coordinates
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Example 3 – pg 1004 # 10

Write the equations in cylindrical coordinates.

a)
3x  2 y  z  6

b)
 x2  y 2  z 2  1
Dr. Erickson
15.8 Triple Integrals in Cylindrical
Coordinates
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Cylindrical Coordinates

Cylindrical 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.
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15.8 Triple Integrals in Cylindrical
Coordinates
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Example 4 – pg 1004 # 12

Sketch the solid described by the given inequalities.
0  

2
rz2
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15.8 Triple Integrals in Cylindrical
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Example 5

Sketch the solid whose volume is given by the integral
and evaluate the integral.
 /2 2 9  r 2
  
0
0
r dz dr d
0
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15.8 Triple Integrals in Cylindrical
Coordinates
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Evaluating Triple Integrals

Suppose that E is a type 1 region whose projection D on
the xy-plane is conveniently described in polar
coordinates.
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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:

E
f ( x, y, z ) dV    
f  x, y, z  dz  dA
u1 ( x , y )


D
u2 ( x , y )
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Evaluating Triple Integrals

However, we also know how to evaluate double
integrals in polar coordinates.
 f  x, y, z  dV
E


 

h2 ( )
h1 ( )
u2  r cos , r sin  

u1 r cos , r sin  
f  r cos  , r sin  , z  r dz dr d
This is formula 4 for triple integration in cylindrical
coordinates.
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Evaluating Triple Integrals

 f  x, y, z  dV   
h2 ( )
h1 ( )
E

u2  r cos , r sin  

u1 r cos , r sin  
f  r cos  , r sin  , z  r dz dr d
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.
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Example 6
Evaluate
z
e
 dV , where E is enclosed by the
E
paraboloid z  1  x 2  y 2 , the cylinder x 2  y 2  5,
and the xy -plane.
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Example 7 – pg. 1004 # 20
Evaluate
 x dV , where E is enclosed by the
E
planes z  0 and z  x  y  5 and by the cylinders
x 2  y 2  4 and x 2  y 2  9.
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Example 8 – pg. 1004 # 27

Evaluate the integral by changing to
cylindrical coordinates.
2
4 y 2
 
2  4 y 2
2

xz dz dx dy
x2  y 2
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Example 9 – pg. 1004 # 31

When 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 continued

Assume 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
Dr. Erickson
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Demonstrations
Feel free to explore these demonstrations
below.
Exploring Cylindrical Coordinates
 Intersection of Two Cylinders

Dr. Erickson
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Coordinates
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