Boise State Math 275 (Ultman) Worksheet 3.5: Triple Integrals in Spherical 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 triple integrals in Cartesian and cylindrical coordinates. Goals In this worksheet, you will: Use the spherical change of coordinate functions to convert expressions in Cartesian coordinates to equations in spherical coordinates. Set up and evaluate triple integrals in spherical coordinates. This includes finding limits of integration, converting the integrand from Cartesian to spherical coordinates, and using the spherical volume element. Set up and evaluate triple integrals that measure volume and mass. Warm-Up: Spherical Coordinates (ρ, φ, θ) x = ρ sin( ) cos( ) y = ρ sin( ) sin( ) z = ρ cos( ) ρ < θ < Identify the angles φ and θ. The parameter ρ / φ / θ is the same as in cylindrical coordinates. Label the spherical radius ρ and the cylindrical radius r. Note: ρ and r are not the same! ρ measures distance from the. r measures distance from the.
Boise State Math 275 (Ultman) Worksheet 3.5: Triple Integrals in Spherical Coordinates Model 1: Cartesian, Cylindrical & Spherical Grid Surfaces Diagram 1A: Grid Surfaces in Cartesian Coordinates ( x, y, z)) < x <, x = constant (y & z vary) < y <, < z < y = constant (x & z vary) z = constant (x & y vary) Diagram 1B: Grid Surfaces in Cylindrical Coordinates (r, θ, z)) r <, r = constant (θ & z vary) ω θ < ω + 2 π, < z < θ = constant (r & z vary) z = constant (r & θ vary) Diagram 1C: Grid Surfaces in Spherical Coordinates (r, θ, z)) ρ <, ρ = constant (φ & θ vary) π, ω θ < ω + 2π φ = constant (ρ & θ vary) θ = constant (ρ & φ vary) 1
Boise State Math 275 (Ultman) Worksheet 3.5: Triple Integrals in Spherical Coordinates 2 Critical Thinking Questions In this section, you will compare grid surfaces in Cartesian, cylindrical, and spherical coordinates. Recall: 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. (Q1) Refer to Diagrams 1A, 1B & 1C: Determine which coordinate system the following are grid surface of, and which variable is held constant. Note: In Diagram 1C, the grid surfaces on the left for ρ = constant are spheres. They are cut open in the diagram to expose the inner spheres. (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: (g) Cones centered about the z-axis: (Q2) Refer to the grid surfaces shown in Diagrams 1A, 1B & 1C: Indicate which coordinate system has grid surfaces that bound (enclose) the following solid (3d) regions. (a) Rectangular boxes, with sides parallel to the xy z-coordinate planes: Cartesian / cylindrical / spherical (b) Solid cylinders centered about the z-axis: Cartesian / cylindrical / spherical (c) Solid balls centered about the origin: Cartesian / cylindrical / spherical (d) Solid cones centered about the z-axis: Cartesian / cylindrical / spherical (We care about these because regions bounded by grid surfaces have constant limits of integration in the coordinate system producing the grid.)
Boise State Math 275 (Ultman) Worksheet 3.5: Triple Integrals in Spherical Coordinates 3 (Q3) In Cartesian coordinates, a rectangular box is a shape with six sides, where opposite sides are parallel, and any two sides that meet make a right angle. In terms of the Cartesian grid, rectangles are regions bounded by Cartesian grid surfaces. In spherical coordinates, we will say that a spherical box is a region bounded by spherical grid surfaces. Sketch three different spherical boxes (refer to Diagram 1C for the bounding grid surfaces): (Q4) The spherical box below has six sides. Label each side indicating which parameter is held constant on that side (ρ = constant, φ = constant, θ = constant). ( Q5) A globe of the earth is a sphere of fixed radius. The lines of latitude are circles parallel to the equator; the lines of longitude are half-circles running from the North Pole to the South Pole. Comparing the lines of latitude and longitude on a globe to spherical coordinates: Lines of latitude correspond to holding φ / θ constant. Lines of longitude correspond to holding φ / θ constant. A difference between lines of latitude on a globe, and the spherical angle φ is: On a globe, the angle representing the North Pole is. In spherical coordinates, the angle φ makes with the positive z-axis (North Pole) is φ =.
Boise State Math 275 (Ultman) Worksheet 3.5: Triple Integrals in Spherical Coordinates 4 ( Q6) The spherical coordinate functions are x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ. In the xy-plane, φ =, so in the xy-plane, cos φ = and sin φ =, and the spherical coordinate functions reduce to: x = y = z = Where have you seen these coordinate functions previously? Interlude: Spherical Volume Element d V Three views of the spherical volume element dv (under the infinite magnifying glass): In spherical coordinates, the volume element is: ( ) ( ) ( ) dv = = = (& etc.) (there are 6 ways of writing dv )
Boise State Math 275 (Ultman) Worksheet 3.5: Triple Integrals in Spherical Coordinates 5 Model 2: Limits of Integration Over Basic Regions Diagram 2A: x 2 + y 2 + z 2 9 Spherical Coordinates ρ θ < Diagram 2B: x 2 + y 2 + z 2 9, x, y Spherical Coordinates ρ θ Diagram 2C: x 2 + y 2 + z 2 9, y, z Spherical Coordinates ρ θ Diagram 2D: 4 x 2 + y 2 + z 2 9, x, y, z Spherical Coordinates ρ θ
Boise State Math 275 (Ultman) Worksheet 3.5: Triple Integrals in Spherical Coordinates 6 Critical Thinking Questions In this section, you will work with triple integrals in cylindrical coordinates for regions bounded by spheres cylindrical grid surfaces. (Q7) In Diagrams 2A, 2B, 2C & 2D: Use the equations in Cartesian coordinates to find the limits for the solid (3d) regions in spherical coordinates. (Fill these in on the tables on the left of the Diagrams.) (Q8) In Diagrams 2A, 2B, 2C & 2D: Label the surfaces bounding the regions and the slices of the region with their equation in spherical coordinates. (The surfaces are in the middle of the Diagrams, the slices are on the right.) (Q9) Match the triple integral W f (ρ, φ, θ) dv in spherical coordinates with its region of integration (one of the regions in Diagrams 2A, 2B, 2C, or 2D). ˆ π/2 ˆ π ˆ 3 ˆ π ˆ π/2 ˆ 3 ˆ π/2 ˆ π/2 ˆ 3 2 ˆ 2π ˆ π ˆ 3 f (ρ, φ, θ) ρ 2 sin φ dρ dφ dθ Region 2 f (ρ, φ, θ) ρ 2 sin φ dρ dφ dθ Region 2 f (ρ, φ, θ) ρ 2 sin φ dρ dφ dθ Region 2 f (ρ, φ, θ) ρ 2 sin φ dρ dφ dθ Region 2 (Q1) Sketch the region W bounded by the spheres x 2 + y 2 + z 2 = 4 and x 2 + y 2 + z 2 = 9, above the xy-plane, for values x. This region can be described by the expressions:. 4 x 2 + y 2 + z 2 9, x, z
Boise State Math 275 (Ultman) Worksheet 3.5: Triple Integrals in Spherical Coordinates 7 (Q11) Write down a triple integral in spherical coordinates that gives the volume of the region W from (Q1). Make sure you use the correct expression for the volume element dv. V W = W ˆ ˆ ˆ dv = d d d ( Q12) Which of the following do not represent the volume of the region W from (Q1)? Circle all that apply. (a) ˆ π/2 ˆ π/2 ˆ 3 π/2 2 dρ dφ dθ (b) ˆ π/2 ˆ π/2 ˆ 3 π/2 ρ 2 sin φ dρ dφ dθ ˆ π/2 ˆ π/2 ˆ 2 π/2 ρ 2 sin φ dρ dφ dθ (c) 2 (d) 1 4 ˆ π/2 ˆ π/2 ˆ 3 2 ˆ 2π ˆ π ˆ 3 2 ρ 2 sin φ dρ dφ dθ ρ 2 sin φ dρ dφ dθ (e) All of the above represent the volume of the region W. ( Q13) Suppose W is the region above the cone φ = π/6 and between the spheres of radius ρ = 1 and z ρ = 3. Set up and evaluate the triple integral dv in spherical coordinates. W x 2 + y 2 + z 2 Make sure you use the correct volume element dv.