# ASU MAT 266 - mat266_ex2_key (5 pages)

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- Pages:
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- School:
- Arizona State University
- Course:
- Mat 266 - Calculus for Engineers II

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MAT 266 Test2 MAT 266 TEST 2 ANSWERS SoMSS ASU Directions 1 There are 14 questions worth a total of 60 points 2 Questions 1 10 are Multiple Choice worth 4 points each to be answered on the supplied SCANTRONS 3 Questions 11 14 are Free Responses worth 5 points each and are to be answered in the space provided on the test 4 Read all the questions carefully 5 For the Free Response you must show all work in order to receive credit 6 When possible box your answer which must be complete organized and exact unless otherwise directed 7 Always indicate how a calculator was used i e sketch graph etc 8 No calculators with QWERTY keyboards or ones like TI 89 or TI 92 that do symbolic algebra may be used Honor Statement By signing below you confirm that you have neither given nor received any unauthorized assistance on this exam This includes any use of a graphing calculator beyond those uses specifically authorized by the Mathematics Department and your instructor Furthermore you agree not to discuss this exam with anyone until the exam testing period is over In addition your calculator s program memory and menus may be checked at any time and cleared by any testing center proctor or Mathematics Department instructor Signature Date 1 MAT 266 Answer d Answer b Test2 1 Find the area of the region bounded by the curves y x2 2 x y x 4 Select the correct answer 125 125 25 a b c 20 d e None of these 3 3 6 2 Find the volume of the solid obtained by rotating the region bounded by y x 2 and x y 2 about the x axis Select the correct answer 6 5 a 3 10 b c 2 5 d 5 e None of these 3 Which of the following integrals is equal to 1 25 Select the correct answer Answer c 1 a 0 Answer c 1 dx x 0 8 1 b 0 1 1 dx x 0 3 c 0 1 dx x 0 2 1 d 0 1 dx x2 e none of these 4 A spring has a natural length of 22 cm If a force of 15 N is required to keep it stretched to a length of 32 cm how much work is required to stretch it from 22 cm to 40 cm Select the correct answer a 3 43 J b 1 93 J c 2 43 J d 3 93 J e None of these 5 Find lim 3 e 2 n n Answer a Select the correct answer a 3 b 0 c 4 d 2 e None of these Answer a 6 A rope 40 ft long weighs 0 8 lb ft and hangs over the edge of a building 110 ft high How much work is done in pulling the rope to the top of the building Select the correct answer a 640 ft lb b 590 ft lb c 641 ft lb d 489 ft lb e None of these Answer b 1 7 Find the sum of the series n 4 3 Select the correct answer 1 1 1 a b c 81 54 3 n d 2 27 e None of these 2 MAT 266 Test2 8 Set up but do not evaluate an integral for the length of the curve y e x sin x 0 x 3 2 Select the correct answer 3 2 Answer b a L 3 2 1 e 2 x 1 sin 2 x dx b L 0 3 2 1 e 2 x 1 sin 2 x dx 0 3 2 1 e 2 x 1 sin 2 x dx c L e None of the above d L 0 1 e 2 x 1 sin 2 x dx 0 9 The integral representing the volume of the solid obtained by rotating the region bounded by the curves y 2 x x 2 y about the y axis is Answer b 2 a 2 y y 2 dy 0 2 b 4 y 2 y 4 dy 0 2 c 2 y y 2 2 y dy 0 4 x x dx 2 0 e None of the above d 2 x 4 10 The integral x 0 1 x dx defines the area between two curves Which of the following 2 integrals calculates the area of the same region using integration with respect to y Answer d Select the correct answer 4 a 2 y 2 y dy 4 b 0 2 2 y y dy 0 2 c y 2 2 y dy 0 2 d 2 y y 2 dy e None of the above 0 3 MAT 266 Test2 FREE RESPONSE 11 The tank shown is full of water Given that water weighs 62 5 lb ft 3 and R 5 ft find the work required to pump the water out of the tank Solution Volume of the ith element 25 xi 2 x Weight of the ith element 62 5 25 xi 2 x Work done to pump out the ith element 62 5 25 xi 2 xi x 5 th Work done to pump out all the water i element 62 5 25 x 2 xdx 9765 625 ft lb 0 12 Evaluate the integral or show that it is divergent 1 x ln x 4 dx 2 Solution 1 First observe x ln x 4 dx u 4 du u ln x 1 3 ln x 3 So t 1 1 dx 4 2 x ln x 4 dx lim t x ln x 2 1 x t x 2 t 3 ln x 3 1 1 1 lim 3 3 t 3 ln t 3 ln 2 3 ln 2 3 lim 4 MAT 266 Test2 13 Set up the integral for the volume of the solid obtained by rotating the region bounded by y x 3 and x y 3 in the first quadrant about the line x 1 Do not evaluate the integral Solution Outer radius of the cross section y1 3 1 Inner radius of the cross section y 3 1 1 Volume y1 3 1 2 y 3 1 2 dy 0 14 The height of a monument is 20 m A horizontal cross section at a distance x meters from the top is an equilateral triangle with side x 4 meters Find the volume of the monument Solution The height of an equilateral triangle with side length equal to s units 3 s 2 1 3 3 2 s s s 2 2 4 3 2 For our problem area of the equilateral triangle x 64 20 3 2 125 3 x dx Hence volume of the monument 64 3 0 So area of the equilateral triangle 5

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