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Exam_exam_3_ - mass of the object RRR D f dV in(a...

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MATH 241, 3rd Examination Prof. Jonathan Rosenberg Monday, November 28, 2005 Instructions. Answer each question on a separate answer sheet . Show all your work. Be sure your name, section number, and problem number are on each answer sheet, and that you have copied and signed the honor pledge on the first answer sheet. The point value of each problem is indicated. The exam is worth a total of 100 points. In problems with multiple parts, whether the parts are related or not, the parts are graded independently of one another. Be sure to go on to subsequent parts even if there is some part you cannot do. Please leave answers such as 5 2 and 3 π in terms of radicals and π and do not convert to decimals . You are allowed use of a non-programmable calculator and one sheet of notes. 1. (40 points, 10 per part) Suppose that an object occupies the solid ball D bounded by the sphere x 2 + y 2 + ( z - 2) 2 = 4 , and has mass density f ( x, y, z ) = z . Set up, but do not evaluate , explicit iterated integrals (with the integrand and all limits of integration written out in the appropriate coordinate system) for the
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Unformatted text preview: mass of the object RRR D f dV in: (a) rectangular (Cartesian) coordinates; (b) cylindrical coordinates; (c) spherical coordinates. (d) Now compute one of the integrals you obtained in (a), (b), or (c). (You will only get credit for evaluating one of the three, though you are free to compute more than one as a way to check your answer. Correct evaluation of an integral that is very far from being correct, in case you made a mistake in (a), (b), or (c), will be worth something, but is not necessarily worth full credit on this part.) 2. (30 points) Let R be the region in the plane bounded by the lines y = x , y = x + 2 , y = 3 x , and y = 3 x + 1 . (See Fgure.) Compute RR R 4 x 2 dA . 1-1-0.5 0.5 1 1.5-3-2-1 1 2 3 4 5 6 3. (30 points) Let Σ = { ( x, y, z ) : x 2 + y 2 + z 2 = 1 } be the unit sphere in 3-space. Compute the surface integral Z Z Σ x 2 dS. 2...
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