MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable CalculusFall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.5. Triple Integrals 5A. Triple integrals in rectangular and cylindrical coordinates 5A-1 Evaluate: a) b) l2 ~d~ lx'2xy2z dz dx dy 5A-2. Follow the three steps in the notes to supply limits for the triple integrals over the following regions of 3-space. a) The rectangular prism having + as its two bases the triangle in the yz-plane cut out by the two axes and the line y z = 1, and the corresponding triangle in the plane x = 1 obtained by adding 1 to the x-coordinate of each point in the first triangle. Supply limits for three different orders of integration: (iii) /// dy dx dz b)* The tetrahedron having its four vertices at the origin, and the points on the three axes where respectively x = 1, y = 2, and z = 2. Use the order /// dz dy dx. c) The quarter of a solid circular cylinder of radius 1 and height 2 lying in the first octant, with its central axis the interval 0 5 y 5 2 on the y-axis, and base the quarter circle in the xz-plane with center at the origin, radius 1,and lying in the first quadrant. Integrate with respect to y first; use suitable cylindrical coordinates. d) The region bounded below by the cone z2 = x2 + y2, and above by the sphere of radius 4 and center at the origin. Use cylindrical coordinates. 5A-3 Find the center of mass of the tetrahedron D in the first octant formed by the coordinate planes and the plane x + y + z = 1. Assume 6 = 1. 5A-4 A solid right circular cone of height h with 90' vertex angle has density at point P numerically equal to the distance from P to the central axis. Choosing the placement of the cone which will give the easiest integral, find a) its mass b) its center of mass 5A-5 An engine part is a solid S in the shape of an Egyptian-type pyramid having height 2 and a square base with diagonal D of length 2. Inside the engine it rotates about D. Set up (but do not evaluate) an iterated integral giving its moment of inertia about D. Assume 6 = 1. (Place S so the positive z axis is its central axis.) 5A-6 Using cylindrical coordinates, find the moment of inertia of a solid hemisphere D of radius a about the central axis perpendicular to the base of D. Assume 6 = 1.. 5A-7 The paraboloid z = x2 + y2 is shaped like a wine-glass, and the plane z = 2x slices off a finite piece D of the region above the paraboloid (i.e., inside the wine-glass). Find the moment of inertia of D about the z-axis, assuming 6 = 1.5B. Triple Integrals in Spherical Coordinates 5B-1Supply limits for iterated integrals in spherical coordinates /Idp dq5 do for each of the following regions. (No integrand is specified; dpdq5d6' is .given "so as to determine the order of integration.) a) The region of 5A-2d: bounded below by the cone z2 = x2 + y2, and above by the sphere of radius fiand center at the origin. b) The first octant. c) That part of the sphere of radius 1and center at z = 1on the z-axis which lies above the plane z = 1. 5B-2 Find the center of mass of a hemisphere of radius a, using spherical coordinates. Assume the density 6 = 1. 5B-3 A solid D is bounded below by a right circular cone whose generators have length a and make an angle 7r/6 with the central axis. It is bounded above by a portion of the sphere of radius a centered at the vertex of the cone. Find its moment of inertia about its central axis, assuming the density 6 at a point is numerically equal to the distance of the point from a plane through the vertex perpendicular to the central axis. 5B-4 Find the average distance of a point in a solid sphere of radius a from a) the center b) a fixed diameter c) a fixed plane through the center 5C. Gravitational Attraction 5C-l.*Find the gravitational attraction of the solid V bounded by a right circular cone of vertex angle 60° and slant height a, surmounted by the cap of a sphere of radius a centered at the vertex of the cone; take the density to be (a) 1 (b) the distance from the vertex. Ans.: a) 7rGa/4 b) 7rGa2/8 5C-2. Find the gravitational attraction of the region bounded above by the plane z = 2 and below by the cone z2 = 4(x2 + y2), on a unit mass at the origin; take 6 = 1. 5C-3.Find the gravitational attraction of a solid sphere of radius 1on a unit point mass Q on its surface, if the density of the sphere at P(x, y, z) is IPQI-'/~. 5C-4.Find the gravitational attraction of the region which is bounded above by the sphere x2 + y2 + z2 = 1and below by the sphere x2 + y2 + z2 = 22, on a unit mass at the origin. (Take 6 = 1.) 5C-5.*Find the gravitational attraction of a solid hemisphere of radius a and density 1 on a unit point mass placed at its pole. Ans: 27rGa(l- d/3)2 E. 18.02 EXERCISES 5C-6.* Let V be a uniform solid sphere of mass M and radius a. Place a unit point mass a distance b from the center of V. Show that the gravitational attraction of V on the point mass is b3a) GM/b2, if b > a; b) GM'/b2, if b 5 a, where M' = -Ma3 Part (a) is Newton's theorem, described in the Remark. Part (b) says that the outer portion of the sphere-the spherical shell of inner radius b and outer radius a -exerts no force on the test mass: all of it comes from the inner sphere of radius b, which has total b3 mass -M .a3 5C-7.* Use Problem 6b to show that if we dig a straight hole through the earth, it takes a point mass m a total of 7rmz 42 minutes'to fall from one end to the other, no matter what the length of the hole is. (Write F = ma, letting x be the distance from the middle of the hole, and obtain an equation of simple harmonic motion for x(t). Here R = earth's radius, M = earth's mass, g = GM/R2
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