18.02 Multivariable Calculus (Spring 2009): Lecture 29Cylindrical and Spherical Coordinates and applicationsApril 17Reading Material: From Simmons: 18.7, 20.6 and 20.7.Last time: Triple Integrals. Cylindrical Coordinates.Today: Cylindrical and Spherical Coordinates. Applications.2 Cylindrical CoordinatesCylindrical coordinates are obtained by using polar coordinates in the xy plane and the z coordinatealong the z-axis, to summarize:rectangular cylindrical = polar + z(x, y, z) (r, θ, z )Conversion: Like polar:(r, θ, z ) −→ (x, y, z)x = r cos θy = r sin θz = z(r, θ, z ) ←− (x, y, z)r =px2+ y2θ = tan−1y/xz = zIntegration in Cylindrical Coordinates:ZZZDf(x, y, z) dV −→ZZZθ r zf(r cos x, r sin x, z)dVz }| {r dz dr dθlimits1Question: WhydV = r dz dr dθ? (2.1)If one wiggles ∆r, ∆θ, ∆z how big is resulting ”cube” ?From the picture is then clear that∆V ≈ (r ∆θ)(∆r)(∆z)hence if we pass to differentials we havedV = r dz dr dθ| {z }typical orderwhich makes sense since we have to take into account the rate of change of the area of the base (withrespect to θ and r) and the rate of change of the hight with respect to z.Question: When do we use cylindrical coordinates? They are very good when the shadow hasrotate symmetry.Exercise 1 (Set up cylindrical integral). Find the mass of the part of paraboloid D described byz = 4 − x2− y2on the upper half space and of density δ(x, y, z) = x + y + z.Solution: The shadow of the paraboloid is the disk c entered at the origin of radius 2. This is aperfect region to describe in polar coordinates, hence we should use the cylindrical coordinates towrite the triple integral representing the mass:23 Spherical CoordinatesThese coordinates are good for- pieces of spheres- hemispheres- ice cream cones.Description of spherical coordinates:Note that here 0 ≤ θ ≤ 2π, 0 ≤ ρ < ∞ and −π ≤ φ ≤ π.Conversion:(ρ, φ, θ) −→ (r, θ, z) −→ (x, y, z)r = ρ sin φ x = r cos θ = ρ sin φ cos θθ = θ y = r sin θ = ρ sin φ sin θz = ρ cos φ z = z = ρ cos φIntegration in spherical coordinate: Given a region D in space31. Find limitsZθ|{z}from shadowZφZρ|{z}from sheetat angle θ2.dV = ρ2sin φ dρ dφ dθ. (3.1)then we haveZZZDf(x, y, z) dV =ZθZρZφf(ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)ρ2sin φ dρ dφ dθQuestion: Why dV = ρ2sin φ dρ dφ dθ?If one wiggles ∆ρ, ∆θ, ∆φ how big is resulting ”cube” ?From the picture is then clear that∆V ≈ (ρ sin φ∆θ)(ρ∆φ)∆ρhence if we pass to differentials we have what we wanted.Exercise 2. Compute the volume of sphere of radius a.Solution: We need to compute the triple integral:ZZZspheredV = volume.4Obviously the best coordinate system is the spherical one.4 The JacobianFollowing the 2D model of change of variables we will call Jacobian of the cylindrical change ofvariable the determinant of the matrix∂x∂θ∂x∂r∂x∂z∂y∂θ∂y∂r∂y∂z∂z∂θ∂z∂r∂x∂z=r sin θ cos θ 0r cos θ sin θ 00 0 1= r,hencedV = r dθ dr dzas in (2.1).On the other hand for the spherical change of variable the Jacobian is given by∂x∂θ∂x∂ρ∂x∂φ∂y∂θ∂y∂ρ∂y∂φ∂z∂θ∂z∂ρ∂x∂φ=−ρ sin φ sin θ sin φ cos θ ρ cos φ cos θρ sin φ cos θ sin φ sin θ ρ cos φ sin θ0 cos φ −ρ sin φ= ρ2sin φhence in this casedV = ρ2sin φ dρ dθ dφ,as in (3.1).5 ApplicationsGiven a solid D in space of density δ(x, y, z) we already com puted the Mass of D asMass(D) =Z Z ZDδ(x, y , z) dV.Now if f(x, y, z) is a function defined on D we define the Average of f to beAverage(f) =R R RDf(x, y, z)δ(x, y, z) dVMass(D).5The Center of Mass of a solid D is the point P = (¯x, ¯y, ¯z), where ¯x is the average of the functionx, ¯y is the average of the function y and ¯z is the average of the function z. Finally we consider theMoment of Inertia. We have• The moment of inertia of a solid D of density δ(x, y, z) about the z-axis:Iz=Z Z ZD(x2+ y2)δ(x, y , z) dV,• The moment of inertia about the x-axis:Ix=Z Z ZD(z2+ y2)δ(x, y , z) dV,• The moment of inertia about the x-axis:Iy=Z Z ZD(z2+ x2)δ(x, y , z) dV.Exercise 3. Compute the moment of inertia Izof the solid cone R between z = ar and z = b. Hereassume δ = 1:6Solution:7Study Guide 1. Consider the following questions:• Consider the cylinder with a disk of radius a as a base and hight h. Using the formula for thevolume given by cylindrical coordinates and by fixing r = a in this formula, compute the areaof the lateral surface of the cylinder. ( Hint: Note that there is no variation dr in this case!)• Consider the sphere of radius a. Using the formula for the volume given by spherical coordinatesand by fixing ρ = a in this formula, compute the area of the sphere. ( Hint: Note that t here isno variation dρ in this
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