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Unformatted text preview: Math 234 Final
Date: May 18th, 2007
Instructor: YOng—Geun Oh No calculator allowed.
Write detailed work to obtain full credits. Your Name: Please circle your TA’s name : Weiyong He Seth Meyer Jie Ling Problem / p
131/40 
P2/30 
P3/20 
P4/20  ‘ P5 /40 
P6/30 
P7/20 
 Total / 200 1. (10 points each) No partial credits for these problems! (a) Change the Cartesian coordinates (x,y,z) :2 ﬁ, — (
dinates (p,t9,¢) With p2 0,0 S 9 < 277, 0 S ¢ 3 77.‘ (b) Evaluate the integrals
1 J17?
/ / ‘ 3m dydm. ’
' o —¢i——zf ‘ (0) Sketch the region of integration in the anyplane. 1 ‘  _
N5, to spher1cal coor (d) Evaluate the iterated integral 2 3m y
/ / / 4dzdyd2:
—1 o m+2 2. Evaluate the following integrals (a) (10 pts) fC(2:c+y) da:+(:r——y) dy ; C' is the line segment from (1,1) to (3,—1). (b) (10 pts)
xdy—ydx_
C $2+y2 .’ C is the unit circle With its center at the origin With counterclockwise orientation. (c) 10 pts) By changing the order of integration, evaluate the integral 4 2 2
f / ey dy dx.
0 z/2 3. (20 pts) Find the volume of the solid bounded by the cylinders 51:2 = y, (2 1)2 2 4g
and the plane y = 3. v O) 4. Consider the function f(:z:, y) = 2x23; — my3 and its graph 2 _= f(m, y). (a) (10 pts) Find the linear approximation of the function at the point (115,34) 2 (1, 1).
(Simplify your answer. N o simpliﬁcation Will result in some points off!) (b) (10 pts) Estimate the maximal possible error for this linear approximation when
the point (:0, y) is chosen so that 1
—1 —1<—. 5. The following problem concerns Green’s formula (a) (10 pts) State precisely Green’s formula for a vector ﬁeld F = Mi + N j on the
plane in circulation—curl form. (You should state clearly about the condi tions on domain of integration and on the orientation of its boundary.
This Will also help you correctly solve the part (c) of this problem!) (10 pts) Choose a suitable vector ﬁeld F and prove using Green’s formula that
if R is a region in the place bounded by a piecewise smooth closed curve C With
the boundary orientation in terms of the region R, then Area ofR = lfctdy ——yda:)
2 c (20 pts) Using the formula found in (b), ﬁnd the area of the region bounded by
the triangle With vertices (1,1), (1, —1) and (2, —2). 6. Consider the surface G in 3space parameterized by
'x(s, t) = (52 + £2, 25, 2t) on the domain given by the disc
82 + t2 S 3 (a) (10 pts) Express the surface element dS in terms of tie, dt. (b) '(20 pts) Suppose the mass density of the surface is given by the function 6(27, y, z) =
3/2 +22. Find the mass of the surface. (Be prepared for doing integration by
substitution you learned in 222. Good substitution will simplify your
calculation.) ' 10 7. Consider the integral (271,0) ’ '
f 3 (yz + y cos 3:) da: + (asz + sin 3:) dy + (my + 2‘) dz. 1213—172) (a) (10 pts) Prove that the integrals do not depend on the choice of paths connecting
the given end points. ' (b) (10 pts) Evaluate the integrals.‘ 12 ...
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 Spring '08
 DICKEY
 Multivariable Calculus

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