tripl_intgrl_sol

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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5. Triple Integrals 5A. Triple integrals in rectangular and cylindrical coordinates 1 Middle: iy + iy2 + yz] 2 = 1 + z - (-1) = 1 + 2z Outer: z + z2] = 6 b) Jd2 Jdfi JdxY y=-1 0 XY ~XY~Z dz dx dy Inner: xy2z2]o = x3y4 Middle: ,x 1 4 y 4]fi = iy6 Outer: kY7] 2 = 0 y. 5A-2 dz dy dx (ii) Jdl Jdl-' Jdl dxdzdy (iii)JdlJdlJdl-'dydXdz c) In cylindrical coordinates, with the polar coordinates r and 8 in xz-plane, 11, we get dy dr de = Jdr" Jdl Jd2dy dr do d) The sphere has equation x2 + y2 + z2 = 2, or r2 + = 2 in cylindrical coordinates. The cone has equation z2 = r2, or z = r. The circle in which they intersect has a radius r found by solving the two equations z = r and z2+r2 = 2 simultaneously; \:I .Y eliminating z we get r2 = 1, so r = 1. Putting it all together, we get cross-sectionview 5A-3 By symmetry, i = y = 2, so it suffices Jdl to calculate Jd just one of these, say 2. We have 1-x 1-z-y z-moment = z dv = z dz dy dx z= 1-X-y 1-x-y 1-2 1nner: iz2] =;(l-~-y)~ Middle:-i(l-~-y)~]~ =i(l-~)~ 0 1 -1 - x4] = I ,, - z - moment. 0 i ; v= mass of D = volume of D = i(base)(height) = - . 1 = i. Therefore 2 = &/+= i; this is also Z and g, by symmetry. 5A-4 Placing the cone as shown, its equation in cylindrical coordinates is z = r and the density is given by 6 = r. By the geometry, its projection onto the xy-plane is the interior R of the origin-centered circle of radius h. vertical cross-section
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TRIPLE INTEGRALS a) Mass of solid D = /~~~6dV=l~~l~~l~~.rdzdrd6 2~h4 Inner: (h - r)r2; Middle: - - - = . Outer: - hr3 3 r41h 4, 12' 12 b) By symmetry, the center of mass is on the z-axis, so we only have to compute its z-coordinate, 5. z-moment of D = //Lz6dv = ~2r~hJlhzT~TdzdTd~ h 1 Inner: iz2r2] = i(h2r2 - r4) Middle: A g) = -h5 . - 2 T 2 (h2: - , 2 15 5A-5 Position S so that its base is in the xy-plane and its diagonal D lies along the x-axis (the y-axis would do equally well). The octants divide S into four tetrahedra, which by
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This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.

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tripl_intgrl_sol - MIT OpenCourseWare http:/ocw.mit.edu...

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