234-borisov-07faex2

234-borisov-07faex2 - Midterm Exam 2, Menday, .Nov. 12,‘...

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Unformatted text preview: Midterm Exam 2, Menday, .Nov. 12,‘ 2007 Lev Borisov, Math 234 DO NOT OPEN THE EXAM BEFORE THE START ANNOUNCEMENT ! Please write, your name and your TA’s name below. Name: . . TA: Each problem is worth 20 points, 'for a total of 100 points. Ca1- culators are not allowed on this test‘ Please read each question carefully, it also helps to check afterwards that you have I answered each part of each question. You must Show all your work to receive credit. When you turn in'your paper after the test, make sure the TA checks your name in their list or writes your name down. Good luck! L ' These are the formulas from section 15.5. You have to know how to write similar formulas for line integrals and double integrals. Here 6 2 6(55, y, z) is a mass density function for a domain D in space. All triple integrals are over the domain D. , Mass and center of mass: 1 M=///w saw/m say—MM , {Moments of inertia about the coordinate axes: Ix=/fl(y2+22)6dV, Iy=///(a:2+22)6dV, Iz=///(a:2+y2)6dV Radii of gyration about the axes: _ Ix _ Ir _ k [1] Sketch thetrevgionof integration and calculate the integral 1/16 1/2 - / cos(167ra:5) dxdy. y=0 =y1/4 Hint: The antiderivative f c0s(167rm5) dIL' can not be expressed in terms of standard functions. * ' [2] Find the centroid of' the quarter of the circle m2+y2 S4, iv 2 0, y 2 0. Remark: By symmetry, the centroid Will lie on the line y = :12. [3] Sketch the region of integration and calculate the integral 1 m 0 / f / V 372 + 3/2 + 22 dzdydx. a:=—1 y=0 z=—-‘/1~z2_y2 I ‘ [4] The curve C in space is the intersection of the cylinder :32 + y2' = 1 with the plane a: + y + z = 10 oriented counterclockwise as viewed from above. . ‘- (a)[10pts] Write a parametric equation of C'. —-——> (b)[10pts] Calculate the work over 0' of the vector field F (:12, y, z) =(y‘1, 0, 2) . P ease eave - ank! [5] Calculate _ /(3a: y2) doc + cos(ey + 1) dy c . . where C’ is the boundary of the triangle‘with vertices (1, —1), (0,0) and (0,2), oriented clOckwise. ...
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This note was uploaded on 09/15/2009 for the course MATH 234 taught by Professor Dickey during the Spring '08 term at Wisconsin.

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234-borisov-07faex2 - Midterm Exam 2, Menday, .Nov. 12,‘...

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