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

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

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 r2 = x2 + y2 tan θ = y/x Dr. Erickson 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. Dr. Erickson 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 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

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 Dr. Erickson 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 Dr. Erickson 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) Dr. Erickson 15.8 Triple Integrals in Cylindrical Coordinates

Example 2 – pg. 1004 # 4 Change from rectangular coordinates to cylindrical coordinates. a) b) Dr. Erickson 15.8 Triple Integrals in Cylindrical Coordinates

Example 3 – pg 1004 # 10 Write the equations in cylindrical coordinates. a) b) Dr. Erickson 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 = c2 is the z-axis. Dr. Erickson 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. Dr. Erickson 15.8 Triple Integrals in Cylindrical Coordinates

Example 4 – pg 1004 # 12 Sketch the solid described by the given inequalities. Dr. Erickson 15.8 Triple Integrals in Cylindrical Coordinates

Example 5 Sketch the solid whose volume is given by the integral and evaluate the integral. Dr. Erickson 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. Dr. Erickson 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: Dr. Erickson 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. Dr. Erickson 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 = 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. Dr. Erickson 15.8 Triple Integrals in Cylindrical Coordinates

Example 6 15.8 Triple Integrals in Cylindrical Coordinates Dr. Erickson 15.8 Triple Integrals in Cylindrical Coordinates

Example 7 – pg. 1004 # 20 Dr. Erickson 15.8 Triple Integrals in Cylindrical Coordinates

Example 8 – pg. 1004 # 27 Evaluate the integral by changing to cylindrical coordinates. Dr. Erickson 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. Dr. Erickson 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? Dr. Erickson 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 Dr. Erickson 15.8 Triple Integrals in Cylindrical Coordinates

Demonstrations Feel free to explore these demonstrations below. Exploring Cylindrical Coordinates Intersection of Two Cylinders Dr. Erickson 15.8 Triple Integrals in Cylindrical Coordinates