234-oh-07spfin-v2

# 234-oh-07spfin-v2 - Math 234 Final Date May 18th 2007...

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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 any-plane. 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 3-space 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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