Calculus II Review Test 2

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Eleanore Strickland
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1 Calculus II Review Test Fid the itegral. ) x( x - 7) 4) - 4x ) x x6 + 8 ) 9 + x 3) x x ) x (l x) 6) 7) 6 - x + (x + ) Evaluate the itegral. cos(l x - 7) ) x 8) x + x4 6) et cot(et - 4 )dt 9) x ( + 4 lx) 7) sec t dt 8) - si x + cos x 9) si x cos x ) ecot v cscv dv ) ( - 6x) e(3x - 9x) ) ) ) x + 8x x + x + 9 -x + 4x + Evaluate the itegral by usig trigoometric idetities ad substitutios to reduce it to stadard form. 3) (csc x + cot x) ) e /t dt t 4) (si x + cos x) 3) 7l x x ) si x sec x

2 6) si x Evaluate the itegral by reducig the improper fractio ad usig a substitutio if ecessary. 8x 7) 8x + 3 8) 4x + 4x x - Evaluate the itegral by separatig the fractio ad usig a substitutio if ecessary. x + 3 9) 6 - x 3) 3) x + x + x x + π/4 cos θ - dθ si θ π/6 Evaluate the itegral by multiplyig by a form of ad usig a substitutio (if ecessary) to reduce it to stadard form. 3) - si x 33) 34) 8 + 8sec x cos x - si x Evaluate the itegral. 3) -3x cos 4x 36) x si x 37) e-x cos 4 x 38) xex 39) 4 9x l x 4) (4x + 7) e-6x Solve the problem. 4) Fid the area betwee y = l x ad the x-axis from x = to x =. 4) Fid the area of the regio eclosed by the curve y = x si x ad the x-axis for 4π x π. 43) Fid the volume of the solid geerated by revolvig the regio i the first quadrat bouded by y = ex ad the x-axis, from x = to x = l 3, about the y-axis. Express the itegrad as a sum of partial fractios ad evaluate the itegral. 4x ) x + 8x + 7 4) 46) 47) 48) x + 7 x + x 4x + 64 x3 + 8x + 6x 7x + x + 96 x3 + 6x 6x + x + 4 (x + )(x - 6)

3 Evaluate the itegral by first performig log divisio o the itegrad ad the writig the proper fractio as a sum of partial fractios. 49) ) 8x3 + 8x + 3 x + x x3 x + x + Evaluate the itegral. ) cos3 4x ) π/4 si7y dy 3) 8 sec4 x 4) π/8 ta4 t dt -π/8 ) si 7x cos 3x 6) cos 9x cos x Solve the problem. 7) Fid the legth of the curve y = l(si x), π/3 x π/ 8) Fid the volume geerated by revolvig the curve y = cos 4x about the x-axis, x π/48 Itegrate the fuctio. 9) - x 6) x 6) 6) 63) t dt x 9x - 6, x > 4x - Evaluate the improper itegral or state that it is diverget. 64) 8x(x + ) 6) 66) 67) 68) (x - ) 9 x3 x e-x6 - xe-x - Evaluate the improper itegral ) 49 - x 7) 7) - 7 x/3 3 x l3x Fid the area or volume. 7) Fid the area of the regio i the first quadrat betwee the curve y = e-x ad the x-axis.

4 73) Fid the volume of the solid geerated by revolvig the regio uder the curve y = x, from x = to x =, about the x-axis. Use partial fractios to fid the sum of the series. 8) (4 - )(4 + 3) 74) Fid the volume of the solid geerated by revolvig the area uder y = 8e-x i the first quadrat about the x-axis 86) 8 ( + 3) Fid the limit of the sequece if it coverges; otherwise idicate divergece. 7) a = l(7-3) - l( - 6) 76) a = 8 + (.) 77) a = ) a = Fid a formula for the th partial sum of the series ad use it to fid the seriesʹ sum if the series coverges. 79) ) (-) - 8) ( + )( + ) +... Fid the sum of the series. 9 8) 4 83) (-)- 7 3 Determie if the series coverges or diverges; if the series coverges, fid its sum. 87) (-) ) e ) 8- Fid the values of x for which the geometric series coverges. 9) 7 x 9) (x - ) 9) x - 8 Fid the sum of the geometric series for those x for which the series coverges. 93) -4 x 84) ) (x - )

5 9) (x + ) Chage the repeatig decimal to a fractio. 96) ) Use the itegral test to determie whether the series coverges. 6 98) 99) ) 6-6 ex - Use the direct compariso test to determie if the series coverges or diverges. ) ) + Use the limit compariso test to determie if the series coverges or diverges. 6 3) 93/ - + Use the root test to determie if the series coverges or diverges. (!) 6) 7) 8) 3 - Determie covergece or divergece of the series ) Determie covergece or divergece of the alteratig series. (-3) ) 87 + ) (-) /3 Determie either absolute covergece, coditioal covergece or divergece for the series. ) (-) ) (-) l Use the ratio test to determie if the series coverges or diverges. 8 4)! 4) (-)- ) 7 7

6 Aswer Key Testame: REVIEW FOR TEST ) l x - 7 ) 9 x / 3) 9 (x 6 + 4) 3/ 4) - 4(l x)4 ) si(l x - 7) 6) l si(et - 4) 7) l sec t + ta t 8) l + cos x 9) si 3x 3 ) - ecot v ) 3 e (3x - 9x) ) - e /t 3) 7 l x l 7 4) si - x ) 3 ta - x 3 6) si- 4 7) ta - (x + ) 8) ta - x 9) ta - ( l x) ) ta - x + 4 ) ta ta - ) si- x - 4 3) -cot x - csc x - x 4) x - cos x ) -cos x 6) x - si x

7 Aswer Key Testame: REVIEW FOR TEST 7) x - 3 l 8x ) x + 3x + 3 l x - 9) x + 3si- x 4 3) x + + l x 3) ) ta x + sec x 33) (cot x - csc x + x) 8 34) l sec x + ta x - l cos x 3) cos 4x - 3 x si 4x 4 36) si x - x cos x 37) 7 38) xex - ex 39) 6.3 4) - 3 x e -6x e -6x 4) l + (-) 4) 9π 43) π(3l 3 - ) (x + )7 44) l (x + 7)3 4) l x7 (x + ) 46) 4 l x x x ) 6 l x + l x ta - x 4 48) 6 l x ta - x 49) 4x - 3 l x l x ) x - x + 3l x + + x + ) si 4x - 6 si 3 4x ) ) 8 ta x ta 3 x 4) π 4-3

8 Aswer Key Testame: REVIEW FOR TEST ) - cos x - 8 cos 4x 6) 4 si 7x + si x 7) l( 3) 8) π 96 + π 3 9) si- 6) 9 l x x 6) ta- 6 6) 4 sec x 63) l x + 4x - 64).63 6) 66) ) 68) Diverget 69) 4 7) 7) 9 l ) 73) π 74) 3π 7) l 7 76) 8 77) e3 78) 79) ) 6 - (-) + 4 8) ( + ) ; 8) 83) 7 4 ; series diverges ;

9 Aswer Key Testame: REVIEW FOR TEST 84) 8) 6 86) ) Coverges; e6 88) Coverges; e6-89) Coverges; 7 9) x < 7 9) < x < 3 9) 6 < x < 93) + 4x 94) 6 - x 9) - x 96) ) 7 98) diverges 99) diverges ) coverges ) Diverges ) Coverges 3) Diverges 4) Coverges ) Coverges 6) Diverges 7) Coverges 8) Coverges 9) Coverges ) Coverges ) Coverges ) Diverges 3) Diverges 4) coverges absolutely

MH1101 AY1617 Sem 2. Question 1. NOT TESTED THIS TIME

MH1101 AY1617 Sem 2. Question 1. NOT TESTED THIS TIME MH AY67 Sem Questio. NOT TESTED THIS TIME ( marks Let R be the regio bouded by the curve y 4x x 3 ad the x axis i the first quadrat (see figure below. Usig the cylidrical shell method, fid the volume of