Is:f(x) dx

12. Speedometer readings for a motorcycle at l2-second intervalsare given in the table.(a) Estimate the distance traveled by the motorcycle duringthis time period using the velocities at the beginning ofthe time intervals.(b) Give another estimate using the velocities at the end ofthe time periods.(c) Are your estimates in parts (a) and (b) upper and lowerestimates? Explain.t (s) 0 12 24 36 48 60v (m/s) 9.1 8.5 7.6 6.7 7.3 8.213. Oil leaked from a tank at a rate of ref) liters per hour. Therate decreased as time passed and values of the rate at twohourtime intervals are shown in the table. Find lower andupper estimates for the total amount of oil that leaked out.t (h) 0 2 4 6 8 10r(t) (L/h) 8.7 7.6 6.8 6.2 5.7 5.314. When we estimate distances from velocity data, it is sometimesnecessary to use times to, t), f2, f3, ... that are notequally spaced. We can still estimate distances using the timeperiods t::.ti = ti - fi-I. For example, on May 7, 1992, thespace shuttle Endeavour was launched on mission STS-49,the purpose of which was to install a new perigee kick motorin an Intelsat communications satellite. The table, providedby NASA, gives the velocity data for the shuttle betweenliftoff and the jettisoning of the solid rocket boosters. Usethese data to estimate the height above the earth's surface ofthe Endeavour, 62 seconds after liftoff.16. The velocity graph of a car accelerating from rest to a speedof 120 km/h over a period of 30 seconds is shown. Estimatethe distance traveled during this period.u--Ih)-----8040/////0 10 20 30 tseco17-19 Use Definition 2 to find an expression for the area underthe graph of f as a limit. Do not evaluate the limit.17. f(x) = $, I ~ x ~ 1618. f(x) = I + x 4 , 2 ~ x ~ 519. f(x) = xcosx, 0 ~ x ~ 7T/220-21 Determine a region whose area is equal to the given limit.Do not evaluate the limit." 2 ( 2i)IO20. lim 2: - 5 + -1/---+00 i=1 n nIl 7T i7T[Q lim 2: - tan-II--HtJ ;=1 4n 4nEvent Time (s) Velocity (m/s)Launch 0 0Begin roll maneuver 10 56End roll maneuver 15 97Throttle to 89% 20 136Throttle to 67% 32 226Throttle to 104% 59 404Maximum dynamic pressure 62 440Solid rocket booster separation 125 126515. The velocity graph of a braking car is shown. Use it to estimatethe distance traveled by the car while the brakes areapplied.u(m/s)156 t(seconds)22. (a) Use Definition 2 to find an expression for the area underthe curve y = x 3 from 0 to 1 as a limit.(b) The following formula for the sum of the cubes of thefirst n integers is proved in Appendix E. Use it to evaluatethe limit in part (a).1 [ n(n+I)]23 + 2 3 + 3 3 + ... + n 3 = 2ITill23. (a) Express the area under the curve y = x 5 from 0 to 2 asa limit.(b) Use a computer algebra system to find the sum in yourexpression from part (a).(c) Evaluate the limit in part (a).ITill24. (a) Express the area under the curve y = x 4 + 5x 2 + x from2 to 7 as a limit.(b) Use a computer algebra system to evaluate the sum inpart (a).(c) Use a computer algebra system to find the exact area byevaluating the limit of the expression in part (b).