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 a t 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 y 2 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 1 Supply limits for iterated integrals in spherical coordinates I d p 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 sphere of radius fi and center at the origin y2 and above by the b The first octant c That part of the sphere of radius 1 and center at z 1 on 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 1 on 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 1 and below by the sphere x2 y2 z2 22 on a unit mass a t 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 E 18 02 EXERCISES 2 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 b3 a GM b2 if b a b GM b2 if b 5 a where M M a3 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 7 r m z 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 M earth s mass R earth s radius g GM R2
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