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. 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

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 = rcosθ y = 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