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- Answers to Even Recommended problems from Ch. 12 Review

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ANSWERS TO EVEN ASSIGNED PROBLEMS CHAPTER 12 REVIEW (pages 723-724) 26. x y z Using the projection on the yz plane we can evaluate the integral as follows: integraldisplayintegraldisplayintegraldisplay E zdV = integraldisplay 1 0 integraldisplay ?1?y2 0 integraldisplay 2?y 0 z dx dz dy = integraldisplay 1 0 integraldisplay ?1?y2 0 (2?y)z dz dy = integraldisplay 1 0 1 2(2?y)(1?y 2)dy = integraldisplay 1 0 1 2(2?y?2y 2 + y3)dy = 13 14 28. Using spherical coordinates we have that 0 ? ? ? 1(since the solid is bounded by the sphere of radius 1), 0 ? ? ? pi2 since the solid lies above the xy plane) and 0 ? ? ? 2pi. Thus integraldisplayintegraldisplayintegraldisplay H z3 radicalbig x2 + y2 + z2 dV = integraldisplay 2pi 0 integraldisplay pi/2 0 integraldisplay 1 0 (?3 cos3 ?)?(?2 sin?) d? d? d? = integraldisplay 2pi 0 d? integraldisplay pi/2 0 cos3 ?sin? d? integraldisplay 1 0 ?6d? = 2pi bracketleftbigg ?14 cos4 ? bracketrightbiggpi/2 0 parenleftbigg1 7 parenrightbigg = pi14 32. x y z The xy projection of the solid is the disk of radius 2 centered at the origin. In cylindrical coordinates the plane y + z = 3 has equation z = 3?rcos?. Thus the volume is given by V = integraldisplayintegraldisplayintegraldisplay E dV = integraldisplay 2pi 0 integraldisplay 2 0 integraldisplay 3?rsin? 0 r dz dr d? = integraldisplay 2pi 0 integraldisplay 2 0 (3r?r2 sin?) dr d? = integraldisplay 2pi 0 [6? 83 sin?] d? = 6?]2pi0 + 0 = 12pi 34. x y z The paraboloid and the half-cone intersect when x2 + y2 =radicalbig x2 + y2, that is when x2 + y2 = 1. Thus the projection of the solid on the xy plane is a circle of radius 1. In cylindrical coordi- nates, the equation of the paraboloid is z = r2 and the equation of the cone is z = r and the volume is given by V = integraldisplayintegraldisplayintegraldisplay E dV = integraldisplay 2pi 0 integraldisplay 1 0 integraldisplay r r2 r dz dr d? = integraldisplay 2pi 0 integraldisplay 1 0 (r2 ?r3) dr d? = integraldisplay 2pi 0 (13 ? 14) d? = 112(2pi) = pi6 42. (a) The surface is a vertical plane at an angle of pi4 with the positive x-axis. In cartesian coordinates, the plane has equation y = x. (b) The surface is a cone at an angle of pi4 radians with the positive z-axis. In cartesian coordinates the cone has equation z = radicalbigx2 + y2. 1 answers_ch12_review.dvi

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