Unit #8 - The Integrl Some problems nd solutions selected or dpted from Hughes-Hllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl distnce the object trveled between t = nd t = 6. 2. The figure below shows the velocity of prticle, in cm/sec, long the t-xis for 3 t 3 (t in seconds). () Describe the motion in words. Is the prticle chnging direction or lwys moving in the sme direction? Is the prticle speeding up or slowing down? (b) Mke over- nd underestimtes of the distnce trveled for 3 t 3. () If we divide the time intervl into n = 4 subintervls, wht is t? Wht re t, t, t 2, t 3, t 4? Wht re f(t ), f(t ), f(t 2 ), f(t 3 ), f(t 4 )? (b) Find the left nd right sums using n = 4. (c) If we divide the time intervl into n = 2 subintervls, wht is t? Wht re t, t, t 2? Wht re f(t ), f(t ), f(t 2 )? (d) Find the left nd right sums using n = 2. 5. At time t, in seconds, your velocity, v, in meters/ second, is given by v(t) = + t 2 for t 6. Use t = 2 to estimte the distnce trveled during this time. Find the upper nd lower estimtes, nd then verge the two. 6. For time, t, in hours, t, bug is crwling t velocity, v, in meters/ hour given by v = + t. Use t =.2 to estimte the distnce tht the bug crwls during this hour. Find n overestimte nd n underestimte. Then verge the two to get new estimte. For questions 7 to, the grph shows the velocity, in cm/sec, of prticle moving long the x-xis. Compute the prticle s chnge in position, left (negtive) or right (positive), between times t = nd t = 5 seconds. 3. Consider the following tble of vlues for f(t). t 5 7 9 2 23 f(t) 3 8 2 3 () If we divide the time intervl into n = 4 subintervls, wht is t? Wht re t, t, t 2, t 3, t 4? Wht re f(t ), f(t ), f(t 2 ), f(t 3 ), f(t 4 )? (b) Find the left nd right sums using n = 4. (c) If we divide the time intervl into n = 2 subintervls, wht is t? Wht re t, t, t 2? Wht re f(t ), f(t ), f(t 2 )? (d) Find the left nd right sums using n = 2. 7. 8. 9. 4. Consider the following tble of vlues for f(t). t 4 8 2 6 f(t) 25 23 22 2 7
.. A cr going 8 ft/s ( bout 9 km/h) brkes to stop in five seconds. Assume the decelertion is constnt. () Grph the velocity ginst time, t, for t 5 seconds. (b) Represent, s n re on the grph, the totl distnce trveled from the time the brkes re pplied until the cr comes to stop. (c) Find this re nd hence the distnce trveled. 2. A 727 jet needs to ttin speed of 32 km/h to tke off. If it cn ccelerte from to 32 km/h in 3 seconds, how long must the runwy be? ( Assume constnt ccelertion.) 3. A student is speeding down Route in his fncy red Porsche when his rdr system wrns him of n obstcle 4 feet hed. He immeditely pplies the brkes, strts to slow down, nd spots skunk in the rod directly hed of him. The blck box in the Porsche records the cr s speed every two seconds, producing the following tble. The speed decreses throughout the seconds it tkes to stop, lthough not necessrily t uniform rte. (b) Give upper nd lower estimtes for the distnce Roger rn in totl during the entire hour nd hlf. (c) How often would Jeff hve needed to mesure Roger s speed in order to find lower nd upper estimtes within. mile of the ctul distnce he rn? 5. The velocity of prticle moving long the x-xis is given by f(t) = 6 2t cm/sec. Use grph of f(t) to find the exct chnge in position of the prticle from time t = to t = 4 seconds. 6. A bsebll thrown directly upwrd t 96 ft/sec hs velocity v(t) = 96 32t ft/ sec t time t seconds. () Grph the velocity from t = to t = 6. (b) When does the bsebll rech the pek of its flight? How high does it go? (c) How high is the bsebll t time t = 5? 7. Two crs strt t the sme time nd trvel in the sme direction long stright rod. The grph below gives the velocity, v, of ech cr s function of time, t. Which cr: () Attins the lrger mximum velocity? (b) Stops first? (c) Trvels frther? Time since brkes pplied (sec) 2 4 6 8 Speed (ft/sec) 8 5 25 () Wht is your best estimte of the totl distnce the student s cr trveled before coming to rest? (b) Which one of the following sttements cn you justify from the informtion given? (i) The cr stopped before getting to the skunk. (ii) The blck box dt is inconclusive. The skunk my or my not hve been hit. (iii) The skunk ws hit by the cr. 4. Roger runs mrthon. His friend Jeff rides behind him on bicycle nd records his speed every 5 minutes. Roger strts out strong, but fter n hour nd hlf he is so exhusted tht he hs to stop. Jeff s dt follow: 8. Two crs trvel in the sme direction long stright rod. The grph below shows the velocity, v, of ech cr t time t. Cr B strts 2 hours fter cr A nd cr B reches mximum velocity of 5 km/ hr. () For pproximtely how long does ech cr trvel? (b) Estimte cr A s mximum velocity. (c) Approximtely how fr does ech cr trvel? Time since Strt (min) 5 3 45 6 75 9 Speed (mph) 2 8 7 () Assuming tht Roger s speed is never incresing, give upper nd lower estimtes for the distnce Roger rn during the first hlf hour. 2
The Definite Integrl 9. The figure below shows Riemnn sum pproximtion with n subdivisions to b f(x)dx. () Is it left- or right-hnd pproximtion? Would the other one be lrger or smller? (b) Wht re, b, n nd x? (b) Wht is 5? 2. Using the figure below, drw rectngles representing ech of the following Riemnn sums for the function f on the intervl t 8. Clculte the vlue of ech sum. () Left-hnd sum with t = 4. (b) Right-hnd sum with t = 4. (c) Left-hnd sum with t = 2. (d) Right-hnd sum with t = 2. 2. The grph of function f(t) is given in the figure below. Which of the following four numbers could be n estimte of f(t) dt, ccurte to two deciml plces? Explin how you chose your nswer. () -98.35 (b) 7.84 (c).2 (d) 93.47 22. () Wht is the re between the grph of f(x) shown below nd the x-xis, between x = nd x = 5? 23. () On sketch of y = ln(x), represent the left Riemnn sum with n = 2 pproximting ln(x) dx. Write out the terms in the sum, but do not evlute it. (b) On nother sketch, represent the right Riemnn sum with n = 2 pproximting ln(x) dx. Write out the terms in the sum, but do not evlute it. (c) Which sum is n overestimte? Which sum is n underestimte? 24. Estimte x 2 dx using left- nd right-hnd sums with four subdivisions, nd then verging them. How fr from the true vlue of the integrl could your finl estimte be? 25. Without computing the sums, find the difference between the right- nd left-hnd Riemnn sums if we use n = 5 subintervls to pproximte 26. Without computtion, decide if π (2x 3 + 4) dx. e x sin x dx is positive or negtive. [Hint: Sketch e x sin(x)]. { x if x 27. () Grph f(x) = x if < x 2 (b) Find the exct vlue of nd see wht shpes you get). (hint: sketch (c) Clculte the 4-term left Riemnn sum pproximtion to the definite integrl. How does the pproximtion compre to the exct vlue? 3
28. Using the figure below, find the vlues of () (c) b c (b) (d) c b c f(x) dx 29. Given the figure below, nd the sttement tht 2 () = 4, estimte (c) The totl shded re. (b) 2 34. Clculte the following pproximtions to () LEFT(2); (b) RIGHT(2); (c) TRAP(2); (d) MID(2) π sin(θ)dθ. 35. Using the tble below, estimte the totl distnce trveled from time t = to time t = 6 using LEFT, RIGHT, nd TRAP. Time, t (s) 2 3 4 5 6 Velocity, v (m/s) 3 4 5 4 7 8 36. For the functions in Problems 36 39, pick which pproximtion- left, right, trpezoid, or midpoint- is gurnteed to give n overestimte for 5, nd which is gurnteed to give n underestimte. (There my be more thn one.) 37. 3. () Using the grph below, find 3. (b) If the re of the shded region is A, estimte 3. 38. 3. Clculte the following pproximtions to () LEFT(2); (b) RIGHT(2); (c) TRAP(2); (d) MID(2) 32. () Find LEFT(2) nd RIGHT(2) for 6 x 2 dx. (x 2 + ) dx. (b) Illustrte your nswers to prt () grphiclly. Is ech pproximtion n underestimte or overestimte? 33. () Find MID(2) nd TRAP(2) for (x 2 + ) dx. (b) Illustrte your nswers to prt () grphiclly. Is ech pproximtion n underestimte or overestimte? 39. 4. () Find the exct vlue of π sin θ dθ without clcultion (i.e. from sketch). 4
(b) Explin, using pictures, why the MID() nd MID(2) pproximtions to this integrl give the exct vlue. (c) Does MID(3) give the exct vlue of this integrl? How bout MID(n)? Explin. 4. The width, in feet, t vrious points long the firwy of hole on golf course is given in the figure below. If one pound of fertilizer covers 2 squre feet, estimte the mount of fertilizer needed to fertilize the firwy. Select the most ccurte estimte pproch from the methods covered in the clss. 5