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.

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Amice Marsh
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1 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 = Produce a solid by revolvig the regio aroud the lie y = -..) Draw the regio beig revolved. 3.) 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 #3 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 = /. (Assume desity = ) 7.) Cosider the regio bouded by y =, y = -, = - ad =. Fid the volume of the solid produced by revolvig the regio aroud = 3. 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 pyramid has a rectagular base with legth, l, ad width, w. Give that l = w, ad give that the height to the peak is 6 feet (ad the peak is directly over the ceter of the base), use cross sectios ad itegratio to fid the volume of the Test # pyramid i terms of w..) A spherical tak has a diameter of feet. Water is i the tak to a depth of seve feet. Suppose water has a desity of 6.3 pouds per cubic foot. How much work is doe i pumpig all of the water out the top of the tak to a poit two feet above the top of the tak?

2 .) For this problem, begi with dp/dt = kp, where P is populatio ad t is time. I 870, the populatio of Arkadelphia was 948. I 880, it was,506. Assumig it was growig epoetially, ad cotiued doig so, what would the populatio of Arkadelphia be i 07? (Note: the actual 06 populatio was 0,793.) 3.) Solve the followig differetial equatio. dy y d 4.) Solve the followig differetial equatio. y" = - si -, y ' (0) =, y (0) = 4 5.) O the aes below, there are marked poits [ote that the origi is marked]. Recogizig that all coordiates are iteded to be itegers, draw the lie segmets for the part of the directio field for each marked poit for the differetial equatio below. y' = + y 6.) Fid the solutio of the iitial-value problem y' + y =, > 0, y() =. 7.) Determie if y = - - is a solutio of the differetial equatio y' + y =. 8.) 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

3 Test #3 thoroughly mied ad drais from the tak at a rate of 5 L/mi. How much salt is i the tak after 0 miutes?.) 4 π/ si 3 d.) ( 3 + si ) d 0 3.) e si d 4.) l d π/ 5.) si 3 cos d 0 6.) sec 6 () ta 6 () d 7.) d ( +4) 8.) d ) Fid the volume of the solid of revolutio obtaied by revolvig the regio bouded by y = si 3, y = 0, = 0, = /6 aroud the lie y = ) 3+ d.) d e Test #4.) Set up BUT DO NOT EVALUATE the partial fractio decompositio for the followig I # ad 3, use partial fractio decompositio to fid the give atiderivative. -.) d 3.) d (+) ( ) I #4-6, evaluate the give limit. 4.) lim ) lim l 0 6.) lim 0 Use the table of itegrals to do the followig. Be sure to idicate which itegratio formula(s) you used. 7.) 3 e d e 9 8.) sec 7 d

4 Test #5 I # ad #, determie covergece or divergece of the give itegrals..) d 4.) / 0 ta d I #3 ad #4, determie covergece or divergece of the give sequeces. 3.) a = (l )/ 4.) a = si (/) 5.) Determie covergece or divergece of the give series. If the series coverges, fid its sum. If it diverges, eplai why. ( 3) 6.) If possible, use the itegral test to determie covergece or divergece for the followig series. I #7 - #9, determie covergece or divergece of the give series. 7.) l 8.) ) cos(/ ) 0.) Use both the limit compariso test ad the basic compariso test to determie covergece or divergece for this series. Test #6 3.) You may assume the followig series coverges by the Alteratig Series Test. a.) Approimate the series usig the first 50 terms. Write dow ALL of the places the calculator gives. b.) Give a upper boud for the error.

5 .) a.) Write the first four o-zero terms of the Maclauri series for si. b.) Use your work from part a.) to approimate si. Use your calculator to simplify your aswer. Write dow ALL of the places the calculator gives. I #3-7, determie absolute covergece, coditioal covergece or divergece for the give series. 3.) k k ( ) k 4.) ( ) 5.) ( ) k k! 3 6.) k ( ) * k 7.) ( ) (* 4*7 *...*(3 )) k k 4k 3 (!) 8.) Fid the radius of covergece for the power series below. 9.) Fid the iterval of covergece for the power series below. Be sure to check the edpoits. ( ) 0.) Use a familiar power series to fid a power series for the followig fuctio. g() = ta - Test #7 =0 =0.) Graph the followig set of parametric equatios. Use mi = -.5, ma =.5, ymi = -, yma =. = cos(3t) y = si(t/) si(t).) Set up the itegral AND USE YOUR CALCULATOR TO EVALUATE THE INTEGRAL to fid the area iside the limaço r = 3 + cos (see graph below)!

6 3.) 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. 4.) 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) 5.) Graph the followig coic. Fill i all blaks. If a blak does ot apply, write "DNA." y - 3 = 0 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 rotated coic. Fill i all the blaks. If a blak does ot apply, write "DNA." 3 y Fill i the blaks. Type of coic Rotated coordiate system Foci Vertices Asymptote Asymptote Ceter Directri Regular coordiate system Verte Verte Asymptote Asymptote

7 7.) Idetify the kid of curve for each of the followig equatios. 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.) h.) i.) A r si A r si A r si j.) + 4y + y + D + Ey + F = 0 k.) + 4y - y + D + Ey + F = 0 l.) + 4y + 6y + D + Ey + F = 0 8.) 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 Coordiates Rectagular Coordiates Focus Focus Verte Verte Asymptote Asymptote Ceter Directri

8 Fial Eam.) A tak has the shape of a iverted right circular coe (larger ed up) with height 0 ft ad radius 4 ft. It is filled with water to a height of 8 ft. Fid the work required to empty the tak by pumpig all of the water to a poit ft above the top of the tak. Oce you set up the itegral, you may use your calculator to evaluate it. (use 6.3 pouds/ft 3 for the desity of water).) Without your calculator, use cylidrical shells to fid the volume of the regio bouded by f() = 5 - ad the -ais whe it is revolved aroud the lie = 5. 3.) Without your calculator, use disks (or washers) to fid the volume if the regio bouded by y =, y = 5 ad y = 5 is revolved aroud the y-ais. d 4.) 5.) 9 d d 6.) 0 7.) e d 8.) si d ) si 3 cos d 0.) d.) NO CALCULATOR!! d.) Cosider the regio bouded by y = - ad the -ais. Write the itegrals ecessary for fidig the -coordiate of the ceter of mass of the regio if the desity is give by () =. Use your calculator to evaluate the itegrals ad give the -coordiate of the ceter of mass. 3.) Solve the followig differetial equatio. y" = - cos -, y(0) =, y'(0) = 4.) Cosider the followig. = cos(3t) y = cos(t/) si(t) a.) b.) Fid a equatio for the taget lie at the poit where t = /. Graph the parametric equatios. 5.) A tak cotais 500L of pure water. Brie that cotais 0. 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 5 miutes?

9 6.) For the followig power series, fid the iterval of covergece. Do ot forget to check the edpoits. 4 7.) Determie covergece or divergece for the sequece e. 8.) Determie covergece (coditioal or absolute) or divergece for the followig series. Be sure to make your reasos clear. 9.) Determie covergece or divergece for the followig series. Be sure to make your reasos clear. l cos 0.) Determie covergece (absolute or coditioal) or divergece for the followig series. Be sure to make your reasos clear.!

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.

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. 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