9.-0 Calculus II exam 6/8/07 All problems are worth 0 poits uless otherwise oted. Show all aalytic work.. (5 poits) Prove that the area eclosed i the circle. f( x) = x +, 0 x. Use the approximate the area bouded by this fuctio ad the x axis. x + y = R is π R. right ed poit ad the midpoit rule for each of rectagles to = 0 x 5. Velocity vt () x 4,. Fid the displacemet ad the distace traveled. 4. Evaluate (x + ) dx. 4 5. Evaluate x ( x ) dx. 6. Evaluate x si( xdx ) 7. Usig PFD, evaluate 8. By a limitig sum, evaluate 9. Evaluate each of the followig: a) (5 poits) b) (5 poits) 0 e π 0 x si dx c) (5 poits) Evaluate x + dx x + 5x + 6 0 x cos xdx d dx x (x + ) dx t l tdt
9. Calculus II exam 7/0/07 All problems are worth 0 poits uless otherwise oted. Show all aalytic work.. Determie whether the improper itegral bouded regio. dx is coverget or ot. If coverget, evaluate it. Graph the x. Fid the area bouded by y = 0 x ad y = x.. Fid the area bouded by x = y ad x = y+. 4. Fid the volume of the solid obtaied by rotatig the lie y = x about the x axis with 0 x 5. 5 5. (5 poits) Use shells to fid the volume of the solid obtaied by rotatig the regio bouded by y = xl x, the x axis, x 4 about the y axis. 6. (5 poits) Prove that the circumferece of a circle is π R. Hit: Assume that x + y = R. =, the x axis ad 0 x. 7. Fid the ceter of mass for the regio bouded by y x 8. (0 poits) Determie whether or ot the followig series are coverget or ot. If coverget, fid the sum aalytically. 4 4 4 a) 4 + + + +... 5 5 5 + b) ( ) = 7 c) = + + d) = +
9. Calculus II exam part II 8/5/07 Name: All problems are worth 5 poits. Show all aalytic work.. By a limitig sum, evaluate. Evaluate x cos(4 xdx ). Evaluate π 4 si ( x)cos ( ) 0 4 0 (x ) dx xdx 4. f( x) = x +, 0 x. Use the bouded by this fuctio ad the x axis. midpoit rule for each of 4 rectagles to approximate the area 5. Fid the area bouded by y = x ad y = x+. 6. Use disks to fid the volume of the solid geerated whe rotatig the regio bouded by y axis about the lie y = 0. = x ad the x 7. Give the geometric series ( ), show that it coverges ad fid its sum. = 5 8. Fid the volume of the solid geerated whe rotatig the regio bouded by x = 4 y ad the y axis about the x = lie. 9. Fid the iterval of covergece for. = 0. Usig the alteratig series test, show that error <.000 ad what is the sum? ( x ) = ( ) 5 is coverget. How may terms are eeded so that
9. Calculus II exam 5/0/ Name: Show all aalytic work ad simplify results. A aswer without work is ot acceptable. Use place accuracy wherever appropriate. There are a total of 06 possible poits.. (0 pts) f( x) = 6x x ad 0 x 6. Use the midpoit rule with 4 rectagles to approximate the area bouded by f(x) ad the x axis. Be sure to set up a table showig all computed values ad graph the fuctio with the approximatig rectagles.. (8 pts) f( x) = x + si( x). Evaluate f ( x) dx whe F(0)=4.5.. (8 pts) f( x) = (6x + x+ ) (6x+ ). Evaluate x= f ( x) dx. x= 0
4. (0 pts) Evaluate + x ( x ) dx 5. (8 pts) d ( ) (cos( ) f x = t dt dx. Evaluate this problem usig the Fudametal Theorem of Calculus, part. Be sure to show all work. x 6. (0 pts) Fid the area bouded by Itegrate with respect to x. y = 5 x ad y x = +. Also graph the bouded regio. Hit:
7. (0 pts) The regio i the first quadrat, bouded by y = x ad x = 4, is revolved about the lie x = -. Fid the volume geerated usig washers. Graph the bouded regio. Hit: Itegrate with respect to y ad simplify results. 8. (8 pts) Use cylidrical shells to fid the volume geerated whe the regio bouded by f( x) = x+ x ad the x axis, 0 x 4, is revolved about the y axis. Graph the bouded regio. 9. (8 pts) Usig disks, fid the volume of the solid geerated whe about the x axis. Graph the bouded regio. y = R x, R x R, is revolved
0. (0 pts) Give that f( x) 4 = x, x, fid the arc legth ad simplify results.. (0 pts) Fid the surface area whe the regio bouded by revolved about the x axis. Graph the bouded regio. y = 5 x ad the x axis, 0 x 5, is. (6 pts) We are uable at this poit to itegrate 0 e x dx. However we are able to approximate the area. Use left ed poits with rectagles ad set up a table showig computed values. Graph the fuctio ad approximatig rectagles.
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9. Calculus II exam 6// Name: Show all aalytic work ad simplify results. A aswer without work is ot acceptable. Use place accuracy wherever appropriate.. (0 pts) A force of 40N is required to hold a sprig that has bee stretched from its atural legth of 0 cm to 5 cm. How much work is doe i stretchig the sprig from 5 cm to 8 cm?. (0 pts) Fid the ceter of mass of the regio bouded by the lie y=x ad the parabola bouded regio. y = x. Also graph the
. (5 pts) Fid the ceter of mass of the regio bouded by y=si(x) ad the x axis, 0 x π. (This problem requires itegratio by parts ad a trig itegral. Graph the bouded regio. 4. a) (5 pts) Evaluate 6 4 ta ( x)sec ( ) x dx b) (5 pts) Evaluate 5 cos ( x)si ( ) x dx
5. (0 pts) y = 5 x with 0 x 5 represets the part of a ellipse i the first quadrat. Fid the area of 5 this part of the ellipse. Graph the bouded regio. 6. (0 pts) Evaluate by trig substitutio + x 4 9 dx
7. (0 pts) Evaluate x 9 dx ( x+ 5)( x ) 8. (0 pts) Use the Trapezoidal rule ad the Simpso s rule to approximate 5 dx x with 4 subdivisios. Set up a table to show all computed values. Also graph the bouded regio ad approximatig trapezoids. Use place accuracy after the decimal poit.
x 9. a) (5 pts) Evaluate the improper itegral dx ad graph the bouded regio. + b) (5 pts) Evaluate the itegral or show that it is diverget. (x + ) dx 0. Determie whether the series is coverget or diverget. If it is coverget, fid its sum. a) (5 pts) = ( ) 5 b) (5 pts) =
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9. Calculus II exam 6/6/ Name: Show all aalytic work ad simplify results. A aswer without work is ot acceptable. Use 4 place accuracy uless otherwise requested.. (0 pts) Evaluate x x e dx usig itegratio by parts.. (0 pts) Fid the volume of the solid obtaied by rotatig the regio bouded by y axis. Graph the bouded regio. y = x, y=8, ad x=0 about the
. (0 pts) Use trigoometric substitutio to evaluate x 9 x dx. 4. (0 pts) Fid the volume of the solid obtaied by rotatig the regio bouded by y = x ad y=. y = x about the lie
5. (0 pts) Give the itegral 5 cos( x) dx x. Use 4 place accuracy for all computatio. a) Use Simpso s Rule to approximate the itegral with =4. Set up a table of values used. b) Use the Midpoit Rule to approximate this itegral with =4. Set up the table of values used. 6. (0 pts) Use series to approximate x cos( x ) dx with a max absolute error < 0.000. 0
+ 7. a) (7 pts) Show that the alteratig series coverges. The fid the partial sum so that the maximum 5 absolute error < 0.0005. = ( ) b) (8 pts) Fid the iterval of covergece for ( ) ad check ed poits. = x
π 8. a) (5 pts) Fid the Taylor polyomial of order 4 geerated by f( x) = si( x) at x =. Also graph si( x ) 4 ad your P 4 approximatio o the iterval x ad idicate the x ad y scales o your graph. b) (0 pts) Use substitutio to develop a polyomial of order 6, i.e. a P 6 approximatio for ( ) =. The determie for what values of x we ca replace the P 6 approximatio with a maximum absolute error < 0.00. f x xe x
9. a) (8 pts) Fid the area eclosed by oe loop of the rose R = cos( θ ). Also graph the rose. b) (7 pts) Give that R = 8cos( θ ), fid the arc legth for oe complete loop of this curve.
Extra work