6.) Find the y-coordinate of the centroid (use your calculator for any integrations) of the region bounded by y = cos x, y = 0, x = - /2 and x = /2.

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Rosamund Alexander
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1 Calculus Test File Sprig 06 Test #.) Fid the eact area betwee the curves f() = 8 - ad g() = +. For # - 5, cosider the regio bouded by the curves y =, y = +. Produce a solid by revolvig the regio aroud the lie y = 9..) Draw the regio beig revolved..) Usig cylidrical shells, SET UP BUT DO NOT EVALUATE the itegral(s) for fidig the volume of the solid above. 4.) Usig disks or washers, SET UP BUT DO NOT EVALUATE the itegral(s) for fidig the volume of the solid above. 5.) Without usig your calculator, evaluate your itegral(s) from # OR #4 (ot both). 6.) Fid the y-coordiate of the cetroid (use your calculator for ay itegratios) of the regio bouded by y = cos, y = 0, = -/ ad = /. 7.) Cosider the regio bouded by y =, y = -, = - ad =. Fid the volume of the solid produced by revolvig the regio aroud =. 8.) Cosider the curve y = si from = to =. a.) Set up but do ot evaluate the itegral for fidig the legth of the curve over that iterval. b.) Set up but do ot evaluate the itegral for fidig the surface area if that curve is revolved aroud y =. c.) Usig your calculator, evaluate the itegral from b.). 9.) A spherical tak has a diameter of 0 feet. It is half full. Suppose water has a desity of 6. pouds per cubic foot. How much work is doe i pumpig all of the water out the top of the tak? Test #.) Cosider the regio, R, bouded by y = si, the -ais, = 0, =. Without usig your calculator, fid the volume produced by revolvig R aroud the -ais..) A flat plate, i the shape of a isosceles trapezoid, is suspeded vertically i a large pool of water. Give a desity of 6.4 pouds per cubic foot, fid the fluid force beig eerted agaist the plate. O the picture, be sure to show how you set up your coordiate system.

2 For problems #-8, fid the give atiderivative. d 4.) ta - d 5.) e cos d.) I #9-0, evaluate the give itegral. Do NOT use your calculator. 0 9.) si d Test # 4 0.) d 4 6.) 4 7.) sec ta d 8.) si 7 cos d I # ad #, determie covergece or divergece of the give itegrals..) d /.) sec d 0.) For this problem, begi with dp/dt = kp, where P is populatio ad t is time. I 880, Moutai Home, Arkasas had a populatio of 7. By 890 it had essetially doubled. Assumig the populatio cotiued to grow epoetially at the same rate, what would the populatio be i 06? (Note: the actual 000 populatio was,0.) Do the followig itegrals. You may ot use your calculator for ay part of these problems. All ecessary work must be show. 4.) d 5.) d (use partial fractio decompositio) ) Set up BUT DO NOT EVALUATE the partial fractio decompositio for the followig ) Solve the followig differetial equatio. dy d y

3 I #8-9, evaluate the give limit. 8.) lim 4 9.) lim si 0 I #0-, use the itegral formulas i your book to work give itegral. l d 6 0.).) csc d 4 9l Test #4.) Determie whether or ot y = + + is a solutio to the differetial equatio below. y " + y ' - 4y = - ( + 4).) Determie covergece or divergece for the give series. If it coverges, fid the sum. Show your work. Give clear reasos for your coclusio. 0.) A tak cotais 000L of pure water. Brie that cotais 0.05 kg of salt per liter of water eters the tak at a rate of 5 L/mi. Brie that cotais 0.04 kg of salt per liter of water eters the tak at a rate of 0 L/mi. The solutio is kept thoroughly mied ad drais from the tak at a rate of 0 L/mi. How much salt is i the tak after 0 miutes? 4.) Solve the followig iitial value problem. y ' = + y I #5-6, determie covergece or divergece for the give sequece. If the sequece coverges, give its limit. Be sure to show all ecessary work. (- ) + si 5.) 6.) e 7.) Let a =, ad a = (8a- + )/, >. a.) Fid a. b.) Fid a. c.) Fid a4. 8.) Determie covergece or divergece for the followig series. If it coverges, fid its sum. ( )

4 9.) Determie covergece or divergece for the give series. Show your work. Give clear reasos for your coclusio. If the series coverges, fid the sum. 0.) Cosider the followig sequece. ta a = a.) Fid a. b.) Fid a. c.) Fid a..) Fid the solutio of the followig differetial equatio. Test #5 dy d = y.) If possible, use the itegral test to determie covergece or divergece for the followig series. Be sure to show whether or ot the itegral test applies. Determie covergece (absolute or coditioal) or divergece for each of the followig series. Make your reasos clear..) 5.) 8.) cos Test #6 l 4.) ( ) l 6.) k ( ) * k k 4k k 4.).) Fid the first three o-zero terms for the Taylor Polyomial for f() = si at c = π..) Use a familiar power series to fid a power series for the followig fuctio. The fid the iterval of covergece. Check the edpoits. f ( ) ( ) 7.)!

5 .) Use the first four terms of the Maclauri series for e to approimate e -. 4.) For the followig power series, fid the iterval of covergece. Do ot forget to check the edpoits. 4 5.) Graph the followig coic ad fill i the blaks. For ay blak that is ot applicable to the particular coic, write "DNA." y Type of coic, Focus, Focus, Edpoit Mior Ais, Edpoit Mior Ais, Verte, Verte, Asymptote, Asymptote, Ceter, Directri, -itercept(s), y-itercept(s) 6.) Graph the followig coic. Fill i all blaks. If a blak does ot apply, write "DNA." y + = 0 Type of coic, Focus, Focus, Edpoit Mior Ais, Edpoit Mior Ais, Verte, Verte, Asymptote, Asymptote, Ceter, Directri, -itercept(s), y-itercept(s) 7.) Graph the followig rotated coic. Fill i all the blaks. If a blak does ot apply, write "DNA." Test #7 y O the et page, fill i the blaks for each coordiate system. Type of coic, Focus, Focus, Edpoit Mior Ais, Edpoit Mior Ais, Verte, Verte, Asymptote, Asymptote, Ceter, Directri, -itercept(s), y-itercept(s).) Fid the area that is iside BOTH of the curves r = ad r = + cos..) Cosider the followig set of parametric equatios. = si t y = cos t a.) Covert the set of equatios to a ice rectagular form (i.e., o trig fuctios). b.) Graph the resultig equatio from part a.). c.) Make ay ecessary chages to your graph from b.) to make it a graph of the parametric equatios. d.) Are the graphs from b.) ad c.) the same? If ot, eplai why ot..) a.) Covert (, -) to polar coordiates. Give the EXACT ANSWER. b.) Covert (-, 5 EXACT ANSWER. 4.) Chage the followig polar fuctio to rectagular form ad graph it. Do ot use your calculator.

6 r = si - cos t 5.) Cosider. y t t a.) Fid a equatio for the taget lie at the poit (4, 6). b.) Fid all values for t that produce a horizotal taget. 6.) Graph the followig. = si(t/) cos(t/) y = cos(t/) si(t) 7.) Graph the followig coic ad fid all of the followig (for each coordiate system) that apply. If oe does ot apply, put "DNA." Give all poits i both polar ad rectagular form. 4 r si Type of coic Polar Focus_ Verte_ Directri -itercept(s) y-itercept(s) GRAPH 8.) Idetify the kid of curve for each of the followig equatios. Rectagular a.) + y + C + Dy + E = 0 b.) + 5y + C + Dy + E = 0 c.) 5-4y + C + Dy + E = 0 d.) y - + Dy + E = 0 e.) C + y + E = 0 f.) - - 5y + C + Dy + E = 0 g.) A r h.) si A r si i.) A r j.) + 4y + y + D + Ey + F = 0 si Fial Eam.) The curve y = si, 0 < < is revolved aroud the -ais. Fid the volume. Do NOT use your calculator.

7 .) Without usig your calculator, fid the eact area betwee the curves f() = - ad g() = +. Evaluate the followig itegrals or atiderivatives..) e d 4.) si cos d 4 5.) d 7.) 4 d 0 d 6.) 8.) e si d 9.) Cosider r = si θ - cos θ. a.) Covert the equatio to rectagular form. b.) Graph the equatio from part a.). 0.) A ball is dropped from a height of 8 feet. Each time it bouces, it goes up half as high as its previous height. What is the total distace traveled by the ball?.) Solve the followig differetial equatio. dy d = y, y(0) = 4 I #-, determie covergece or divergece for the give sequece. If the sequece coverges, give its limit. Be sure to show all ecessary work. (- ) + si.).) e 4.) Determie covergece or divergece for the give series. Show your work. Give clear reasos for your coclusio. If the series coverges, fid the sum. ta 5.) If possible, use the itegral test to determie covergece or divergece for the followig series. l 6.) Determie covergece or divergece for the followig series. Be sure to make your reasos clear. l k k k 7.) Determie covergece or divergece for the give series. If it coverges, determie if the covergece is absolute or coditioal. Be sure to make your reasos clear.

8 8.) For the followig power series, fid the iterval of covergece. Do ot forget to check the edpoits. 0 / 0 9.) Graph the followig. Use mi = ymi = -0, ma = yma = 0 = cos t - 6 cos(t/6) y = si t - 6 si(t/6) 0.) Graph the followig coic ad fid all of the followig (for each coordiate system) that apply. If oe does ot apply, put "DNA." + y = 0 Type of coic Focus Verte Directri -itercept(s) y-itercept(s) GRAPH

7.) Consider the region bounded by y = x 2, y = x - 1, x = -1 and x = 1. Find the volume of the solid produced by revolving the region around x = 3.

7.) Consider the region bounded by y = x 2, y = x - 1, x = -1 and x = 1. Find the volume of the solid produced by revolving the region around x = 3. Calculus Eam File Fall 07 Test #.) Fid the eact area betwee the curves f() = 8 - ad g() = +. For # - 5, cosider the regio bouded by the curves y =, y = 3 + 4. Produce a solid by revolvig the regio aroud