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Chapter 15 – Multiple Integrals. 15.8 Triple Integrals in Cylindrical Coordinates. Objectives: Use cylindrical coordinates to solve triple integrals. 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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Chapter 15 – Multiple Integrals

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## Chapter 15 – Multiple Integrals

15.8 Triple Integrals in Cylindrical Coordinates

• Objectives:

• Use cylindrical coordinates to solve triple integrals

15.8 Triple Integrals in Cylindrical Coordinates

### Polar Coordinates

• In plane geometry, the polar coordinate system is used to give a convenient description of certain curves and regions.

15.8 Triple Integrals in Cylindrical Coordinates

### 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), thenx = rcosθy = r sin θ r2 = x2 + y2 tan θ = y/x

15.8 Triple Integrals in Cylindrical Coordinates

### 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.

15.8 Triple Integrals in Cylindrical Coordinates

### 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 Ponto the xy–plane.

• z is the directed

distance from the

xy-plane to P.

15.8 Triple Integrals in Cylindrical Coordinates

### Cylindrical Coordinates

• To convert from cylindrical to rectangular coordinates, we use the following (Equation 1):

x = rcosθ

y = r sin θ

z = z

15.8 Triple Integrals in Cylindrical Coordinates

### Cylindrical Coordinates

• To convert from rectangular to cylindrical coordinates, we use the following (Equation 2):

r2 = x2 + y2

tan θ = y/x

z = z

15.8 Triple Integrals in Cylindrical Coordinates

### Example 1

• Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point.

• a)

• b)

15.8 Triple Integrals in Cylindrical Coordinates

### Example 2 – pg. 1004 # 4

• Change from rectangular coordinates to cylindrical coordinates.

• a)

• b)

15.8 Triple Integrals in Cylindrical Coordinates

### Example 3 – pg 1004 # 10

• Write the equations in cylindrical coordinates.

• a)

• b)

15.8 Triple Integrals in Cylindrical Coordinates

### 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 = c2is the z-axis.

15.8 Triple Integrals in Cylindrical Coordinates

### Cylindrical Coordinates

• In cylindrical coordinates, this cylinder has the very simple equation r = c.

• This is the reason for the name “cylindrical” coordinates.

15.8 Triple Integrals in Cylindrical Coordinates

### Example 4 – pg 1004 # 12

• Sketch the solid described by the given inequalities.

15.8 Triple Integrals in Cylindrical Coordinates

### Example 5

• Sketch the solid whose volume is given by the integral and evaluate the integral.

15.8 Triple Integrals in Cylindrical Coordinates

### Evaluating Triple Integrals

• Suppose that E is a type 1 region whose projection D on the xy-plane is conveniently described in polar coordinates.

15.8 Triple Integrals in Cylindrical Coordinates

### 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:

15.8 Triple Integrals in Cylindrical Coordinates

### Evaluating Triple Integrals

• However, we also know how to evaluate double integrals in polar coordinates.

• This is formula 4 for triple integration in cylindrical coordinates.

15.8 Triple Integrals in Cylindrical Coordinates

### Evaluating Triple Integrals

• It says that we convert a triple integral from rectangular to cylindrical coordinates by:

• Writing x = rcosθ, y = r sin θ.

• Leaving z as it is.

• Using the appropriate limits of integration for z, r, and θ.

• Replacing dV by r dzdr dθ.

15.8 Triple Integrals in Cylindrical Coordinates

### Example 6

15.8 Triple Integrals in Cylindrical Coordinates

### Example 7 – pg. 1004 # 20

15.8 Triple Integrals in Cylindrical Coordinates

### Example 8 – pg. 1004 # 27

• Evaluate the integral by changing to cylindrical coordinates.

15.8 Triple Integrals in Cylindrical Coordinates

### 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.

15.8 Triple Integrals in Cylindrical Coordinates

### 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?

15.8 Triple Integrals in Cylindrical Coordinates

### 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

15.8 Triple Integrals in Cylindrical Coordinates

### Demonstrations

Feel free to explore these demonstrations below.

• Exploring Cylindrical Coordinates

• Intersection of Two Cylinders

15.8 Triple Integrals in Cylindrical Coordinates