18.02A Topic 42: Triple integrals; rectangular and cylindrical coordinates.Author: Jeremy OrloffRead: SN: I.3, TB: 20.5, 20.6Ingredients: w = f(x, y, z) a funcion of 3 variables.D = a volume in space.Examples:Box Cylinder SphereThe triple integral is the usual sum over all the volume elements making up D:ZZZDf(x, y, z) dV , where dV = dx dy dz.xyzdydxdzEven more than in two dimensions the hard part is finding the limits of integration.The rest of the hour will be examples.Example 1: Find the average distance of a point in the tetrahedronshown to the xy-plane, i.e., compute Ave =1VZZZDf(x, y, z) dV .xyz111RLimits: we’ll use vertical line segments to fill out the volume.z goes from 0 to the slanted plane.x, y are in R = the projection of D to the xy-plane.⇒ inner variable z: 0 to 1 − x − yR(middle variable y : 0 to 1 − xouter variable x : 0 to 1xyx + y = 111Rf(x, y, z) = distance to xy-plane = z.Volume of tetrahedron = 1/6 ⇒ ave = 6Z10Z1−x0Z1−x−y0z dz dy dx.Inner:z221−x−y0=(1 − x − y)22.Middle:Z1−x0(1 − x − y)22dy = −(1 − x − y)361−x0=(1 − x)36.Outer: 6−(1 − x)42410=14.118.02A topic 42 2Example 2: For a cylinder of height h, radius a and density δ = 1find the moment of inertia about the central axis.I =RRRDr2dm =RRRDr2· δ dV .Limits: Inner z: 0 to h.x, y in R which is a disk of radius a.Inner z: 0 to hMiddle r: 0 to aOuter θ: 0 to 2π.⇒ I =Z2π0Za0Zh0r2dz r dr dθ.Inner integral: r3zh0= hr3.Middle integral:Za0r3h dr =a4h4.Outer integral: 2πa44h = Ma22.zyxExample 3: Find the volume between the paraboloid z = x2+ y2and the plane z = 2yHow to find R? Pictures if possible, sometimes need algebra.C0= projection of intersection of the surfaces.That is, C0is given by x2+ y2= 2y ⇔ x2+ (y − 1)2= 1 –a circleLimits: inner z: from x2+ y2to 2y (the yz section shown at rightshows the plane is above the parabola)middle r: from 0 to 2 sin θouter θ: from 0 to πSince we’re using polar coordinates we need to rephrase the inner limits:Inner z: from r2to 2r sin θ.⇒ V =Zπ0Z2 sin θ0Z2r sin θr2r dz dr dθ =π2.xyzRC0Example 3: graphyzz = y2z = 2yExample 3: section in yz-planexy1C0Example 3: curve C0We call x, y, z rectangular coordinates.We call r, θ, z cylindrical coordinates.xyzdxdydzdxdyVol. element in rectangular coord.xyzr dθdrdzdθVol. element in cylindrical coord.18.02A topic 42 3Example 4: Find the volume between the paraboloids z = 4 − x2− y2and z = x2+ y2.R = projection of volume to the xy-plane.C0= projection of the intersection of the surfaces,i.e., x2+ y2= 4 − x2− y2⇒ x2+ y2= 2.That is, C0is a circle of radius√2 centered at 0.Limits: inner z: x2+ y2to 4 − x2− y2⇔ r2to 4 − r2.middle r: 0 to√2.outer θ: 0 to 2π.⇒ Volume =Z2π0Z√20Z4−r2r2r dz dr dθ. (Easy to com-pute.)xyzRC1z = x2+ y2z = 4 − x2− y2Part of volume in first octantExample 5: Use cylindrical coordinates to find the center of mass of the hemisphereshown. (Assume δ = 1.)By symmetry it’s clear xcm= 0 and ycm= 0.zcm=1MZZZDz dm =1MZZZDz δ dV .Clearly R is a disc of radius 2 and M =163π.Limits: inner z: from 0 top4 − x2− y2=√4 − r2.middle r: from 0 to 2.outer θ: from 0 to 2π.⇒ zcm=316πZ2π0Z20Z√4−r20zr dz dr dθ.Inner:316πz2r2√4−r20=316π4 − r22· r =316π4r − r32.Middle:316πr2−r4820=38π.Outer:38π2π =34⇒ zcm=34.yzx2x2+ y2+ z2= 4End of topic 42
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