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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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 Spring '08
 DICKEY
 Multivariable Calculus

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