10sE3ics - Prof Girardi Math 142 Spring 2010 04.19.10 Exam...

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Unformatted text preview: Prof. Girardi Math 142 Spring 2010 04.19.10 Exam 3 - inclass part MARK BOX INSTRUCTIONS: (1) To receive credit you must: (a) work in a logical fashion, show all your work, indicate your reasoning; no credit will be given for an answer that just appears; such explanations help with partial credit (b) if a line/ box is provided, then: — show you work BELOW the line/ box — put your answer on/ in the line/ box (0) if no such line/ box is provided, then box your answer (2) The MARK BOX indicates the problems along with their points. Check that your copy of the exam has all of the problems. (3) You may not use a calculator, books, personal notes. (4) During this exam, do not leave your seat. If you have a question, raise your hand. When you finish: turn your exam over, put your pencil down, and raise your hand. (5) This exam covers (from Calculus by Stewart, 6th ed., ET): take home part 11.9—11.11 and inclass part 6.1—6.3 . Problem Inspiration: Mostly homework and old exam problems. See the solution key for details. Honor Code Statement I understand that it is the responsibility of every member of the Carolina community to uphold and maintain the University of South Carolina’s Honor Code. As a Carolinian, I certify that I have neither given nor received unauthorized aid on this exam. Furthermore, I have not only read but will also follow the above Instructions. Signature : 1 & 2. Fill—in—the—blanks/boxes. o In 1a and 2a, fill in the blank with: perpendicular or parallel. o In 1b, 1c, 1d, 2b, 2c, fill in the blank with a formula involving some of : 2 , 7r , radius , radiusbig , radiuslittle , average radius , height , and/or thickness. 1- Disk/Washer Method Let’s say you revolve some region in thew-plane around an axis of revolution so you get a solid of revolution. Next you Want to find the volume of this solid of revolution using the disk or washer method. 1a. You should partition the coordinate axis (3.9., the. x—axis or the y—axis) that is to the axis of revolutibn. 113- If you use the disk method, then the volume of a typical disk is: 7T '(‘rmclius)z (l2)?! 16., If you partition the z-axis, the Az & . 2. Shell Method Let’s say you molve some region in the xyuplane around an, axis of revolution so you get a solid of revolution; Next you wam to find the vomm‘e of this solid of. revolution using the shell method. i 23. You should partition the coordinate (Le, them-axis or the y~axis) that is pr «‘3le r to the axis of revolution. 2b. If you use the Shell method, then the volume of atypical shell is; ’llm'chness "” 26¢ If you partition the z-axis, the A2 2 3. Let R be the region enclosed by y = m and y = a: + 2 . Let A be the area of the region R. , 3a. The points ofintersection ofy = m2 and y = a3+2 are P =( 'l , l ) and Q =( 2 , l Make a rough sketch of the region R, labeling P and Q. 1 :7” 1R 4%), 151—2: 0 {m3 (ac—alum") 90 @553»: 1," 3b. Express the area A as integral(s) With respect to ac (so you want daj). 30. Express the area A as integral(s) with respect to y (so you want dy). You do NOT have to evaluate the integral(s) nor do lots of algebra. 4. Sketched below is the region R that is enclosed by y=3ac2 and y=0 and 23:1 and m=2. 4a. In the sketch below, draw in a typical rectangle (should it be horizontal that would be used to express the area of R as precisely 1 integral (and not 2 integrals . \ l (2, IL) [L r. In each of problems 4b, 4c, 4d: 0 R will be revolved around some line to create a solid of revolution 0 using either the disk, washer, or shell method7 express the volume V of the resulting solid of revolution as one integral (and NOT as 2 or more integrals). o In the space provided below each problem, make some good enough sketch (does not have to be too fancy) to indicate (i.e., help justify) your thinking/reseasoning behind your solution 0 you do not have to do lots of algebra to your integrand A c you do not have to integrate your integral. 4b. The volume V of the solid obtained b re'volvin the re 'on R about the mantis is ' w) 3} 5.. . <3 V ..,- ,1!» ‘9 2»?th ’X' 023553 Disk”: Rewind: . M Via lilTWsA‘: of +693 L'zmzmrmf 40- The volume V of the solid obtained by revoiving the region R about the y-axis is §e w , 1’1 V, S amiwwx 2 m" E 953444 E? X35 1 :r .- TE . “’2? {:2 ' Par+i1réav1 wi$fié :x m}; {gacfi i ' \ = 211 a934, (Jéwéws) : : mmcax‘) ( A“) z a . a s 7 Késiéiéa'eéfi 52H: —- X’Numfi '=*"‘ ‘€‘ 2'? ‘3 c w — w W : WK 0} * [IZ’Bx‘flzlAX / m we 5. Using th/was‘éér method, express as an integral (do not evaluate) the volume of a frustum of a. right circular cone with hei ht h, lower base radius R, and top radius r. A frustum of a right circular cone with height [1, lower base radius R, and top radius r \V/CALW; OJ ( «flee. +uma~ cram) / m Ma 5. Using them method, express as sphere with radius r and height h. an integral (do not evaluate) the volume of a cap of a ...
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10sE3ics - Prof Girardi Math 142 Spring 2010 04.19.10 Exam...

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