Worksheet 3.4: Triple Integrals in Cylindrical Coordinates. Warm-Up: Cylindrical Volume Element d V

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1 Boise State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates From the Toolbox (what you need from previous classes) Know what the volume element dv represents. Be able to find limits of integration for double integrals in polar coordinates, and triple integrals in Cartesian coordinates. Goals In this worksheet, you will: Use the cylindrical change of coordinate functions to convert expressions in Cartesian coordinates to equations in cylindrical coordinates. Set up and evaluate triple integrals in cylindrical coordinates. This includes finding limits of integration, converting the integrand from Cartesian to cylindrical coordinates, and using the cylindrical volume element. Warm-Up: Cylindrical Volume Element d V The volume element d V in cylindrical coordinates is the volume of an infinitesimal box, where the base of the box is the polar area element da, and the height of the box is an infinitesimal change in z. Under the infinite magnifying glass: So, the cylindrical volume element is: dv = da polar dz = d d dz

2 Bosie State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates 1 Model 1: Grid Surfaces in Cartesian & Cylindrical Coordinates Diagram 1A: Grid Surfaces in Cartesian Coordinates (x, y, z ) < x <, < y <, < z < x = (constant) (y & z vary) y = (constant) (x & z vary) z = (constant) (x & y vary) Diagram 1B: Grid Surfaces in Cylindrical Coordinates (r, θ, z ) r <, ω θ < ω + 2π, < z < r = (constant) (θ & z vary) θ = (constant) (r & z vary) z = (constant) (r & θ vary) Critical Thinking Questions In this section, you will compare grid surfaces in Cartesian and cylindrical coordinates, and compare polar and cylindrical coordinates. A grid surface of a 3-d coordinate system is a surface generated by holding one of the coordinates constant while letting the other two vary.

3 Bosie State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates 2 (Q1) Refer to Diagrams 1A & 1B: Determine whether the following surfaces are grid surfaces in either Cartesian or cylindrical coordinates. If they are, indicate for which coordinate system(s), and which coordinate is held constant. (a) Planes parallel to the y z-plane: (b) Planes parallel to the xz-plane: (c) Planes parallel to the xy-plane: (d) Cylinders centered about the z-axis: (e) Half-planes perpendicular to the xy-plane: (f) Spheres centered about the origin: (Q2) Refer to Diagrams 1A & 1B: Which of the following solid 3-d regions are bounded by the grid surfaces of either the Cartesian or cylindrical coordinate system? (a) Rectangular boxes, with sides parallel to the xy z-coordinate planes: Cartesian coordinate system / cylindrical coordinate system / neither (b) Solid cylinders centered about the z-axis: Cartesian coordinate system / cylindrical coordinate system / neither (c) Solid balls centered about the origin: Cartesian coordinate system / cylindrical coordinate system / neither (Q3) We will say that a cylindrical box is a region bounded by pairs of cylindrical grid surfaces r = c 1, r = c 2, θ = k 1, θ = k 2, z = d 1, z = d 2 (for c 1, c 2, k 1, k 2, d 1, d 2 constants). Sketch three different cylindrical boxes (refer to Diagram 1B for the bounding grid surfaces):

4 Bosie State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates 3 (Q4) The diagrams below represents a plane parallel to the xy-plane; in fact, this is a grid surface for cylindrical coordinates, z = (constant) (shown on the right of diagram 1B). (a) On the left, sketch the curves of intersection for one of the planes z = (constant) with the set of grid surfaces r = (constant) (shown on the left of Diagram 1B). (b) In the middle, sketch the curves of intersection for one of the planes z = (constant) with the set of grid surfaces θ = (constant) (shown on the middle of Diagram 1B). (c) On the right, combine the curves of intersection from parts (a) and (b). Intersection of z = c & grid surfaces r = (constant). Intersection of z = c & grid surfaces θ = (constant). Intersection of z = c & grid surfaces r, θ = (constant). (Q5) The curves of intersection you sketched in (Q4) should look familiar. Where have you seen them before? ( Q6) The cylindrical coordinate functions x = r cos θ, y = r sin θ, z = z lead to the grid surfaces you ve been working with so far. In particular, when r = (constant), the grid surfaces are cylinders centered about the z-axis. (a) Find the three coordinate functions so that the cylinders are centered about the x-axis. x = y = z = (b) Find the three coordinate functions so that the cylinders are centered about the y-axis. x = y = z =

5 Bosie State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates 4 Model 2: Integrating Over Basic Regions Diagram 2A Cartesian Coordinates Cylindrical Coordinates x 2 + y 2 4 r θ z 3 z Diagram 2B Cartesian Coordinates Cylindrical Coordinates 1 x 2 + y 2 4 r θ z 3 z Diagram 2C Cartesian Coordinates Cylindrical Coordinates 1 x 2 + y 2 4 r x, y θ z 3 z Critical Thinking Questions In this section, you will work with triple integrals in cylindrical coordinates for regions bounded by cylindrical grid surfaces. (Q7) In Diagrams 2A, 2B & 2C: Use the equations in Cartesian coordinates to find the equations for the bounding surfaces in cylindrical coordinates. Label the bounding surfaces in the three diagrams with their equations in cylindrical coordinates.

6 Bosie State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates 5 (Q8) Use the limits of integration to match the triple integrals below with its region of integration (one of the regions in Diagrams 2A, 2B, or 2C). ˆ 3 ˆ 2π ˆ 2 1 ˆ 3 ˆ 2π ˆ 2 ˆ 3 ˆ π/2 ˆ 2 (r + z) r dr dθ dz Region from Diagram 2 1 ˆ π/2 ˆ 2 ˆ 3 1 5z r dr dθ dz Region from Diagram 2 z sin θ r dr dθ dz Region from Diagram 2 z 2 r dz dr dθ Region from Diagram 2 (Q9) Sketch the solid cylinder W bounded by the cylinder x 2 + y 2 = R 2, between the xy-plane and the plane z = H. (Q1) Write down a triple integral in cylindrical coordinates that gives the volume of the solid cylinder W from (Q9). Make sure you use the cylindrical volume element dv that you found in the Warm-Up! Then, evaluate the integral to show that the volume of this region is V W = πr 2 H. V W = W 1 dv = W ˆ ˆ ˆ dv = d d d

7 Bosie State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates 6 ( Q11) Which of the following represent the volume of the solid cylinder W from (Q9)? Circle all correct answers, then fix the incorrect answers. (a) ˆ 2π ˆ H ˆ R dr dz dθ (b) ˆ 2π ˆ H ˆ R ˆ π (c) 2 (d) ˆ H ˆ R ˆ 2π ˆ R ˆ H ˆ 2π (e) 2 ˆ H/2 ˆ R R dr dz dθ r dr dz dθ r dr dz dθ r dr dz dθ (f) None of the above represent the volume of the region W. ( Q12) Suppose W is the solid region bounded by the cylinder x 2 + y 2 = 4 and the planes z = and z = x + y + 3. Set up a triple integral in cylindrical coordinates that gives the volume of W. ( Q13) Suppose W is the solid cylinder bounded by the cylinder x 2 + y 2 = 25 and the planes z = and z = 3. Find the value of the radius r that separates the solid cylinder into two regions of equal volume.

8 Bosie State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates 7 Model 3: Integrating Over Other Regions Diagram 3A: Region inside the cone z = x 2 + y 2 and below the plane z = 3 In cylindrical coordinates, the equations of the cone and plane are: cone: z = r plane: z = 3 Diagram 3B: Region inside the sphere x 2 + y 2 + z 2 = 9 In cylindrical coordinates, the equation of the sphere is: sphere: r 2 + z 2 = 9 Critical Thinking Questions In this section, you will work with triple integrals in cylindrical coordinates for regions bounded by surfaces that are not grid surfaces.

9 Bosie State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates 8 (Q14) Using the sketches in Diagrams 3A and 3B, match the given limits of integration for integrals over the cone in Diagram 3A and the sphere in Diagram 3B with the correct region of and the volume element giving the correct order of integration: (a) Limits: θ 2π, z 3, r z or: ˆ 2π ˆ 3 ˆ z Region of Integration: Cone Sphere Order of Integration: dv = r dz dr dθ dv = r dr dz dθ (b) Limits: θ 2π, 3 z 3, r 9 z 2 or: ˆ 2π ˆ 3 ˆ 9 z 2 3 Region of Integration: Cone Sphere Order of Integration: dv = r dz dr dθ dv = r dr dz dθ (c) Limits: θ 2π, r 3, r z 3 or: ˆ 2π ˆ 3 ˆ 3 r Region of Integration: Cone Sphere Order of Integration: dv = r dz dr dθ dv = r dr dz dθ (d) Limits: θ 2π, r 3, 9 r 2 z 9 r 2 or: ˆ 2π ˆ 3 ˆ 9 r 2 9 r 2 Region of Integration: Cone Sphere Order of Integration: dv = r dz dr dθ dv = r dr dz dθ (Q15) Using your answers from (Q14), set up (but do not evaluate) the triple integral in cylindrical coordinates that gives the volume of the sphere x 2 + y 2 + z 2 = 9, using the given order of integration: (a) dv = r dz dr dθ: (b) dv = r dr dz dθ:

10 Bosie State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates 9 (Q16) Using your answers from (Q14), set up (but do not evaluate) the triple integral in cylindrical coordinates that gives the mass of a solid region with density δ(x, y, z) = (x 2 + y 2 + z)g/cm 3, bounded by the cone z = x 2 + y 2 and the plane z = 3, using the given order of integration: (a) dv = r dz dr dθ: (b) dv = r dr dz dθ: ( Q17) Set up and evaluate a triple integral in cylindrical coordinates that gives the volume of the napkin ring formed by drilling a cylinder x 2 + y 2 = R 2 out of a sphere x 2 + y 2 + z 2 = P 2 (where R and P are constants with R P ).