Module 14: THE INTEGRAL Exploring Calculus


 Lenard Francis
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1 Module 14: THE INTEGRAL Explorng Calculus Part I Approxmatons and the Defnte Integral It was known n the 1600s before the calculus was developed that the area of an rregularly shaped regon could be approxmated by dvdng the regon nto small peces and computng the total area of the peces. By choosng more and more peces, the error between the total area of the peces and the area of the regon became less and less. We credt the ancent Greeks wth the dscovery of ths method. Before 1850 mathematcans, begnnng wth Newton and Lebnz, understood the connecton between the ntegral and the area under a curve. The defnton of the defnte ntegral used today, whch expresses the area of the regon between a certan class of functons and the xaxs, as the lmt of a specally selected approxmatng sum, s attrbuted to the great mathematcan Fredrck Remann, c In ths module, we wll dscover more about the relatonshp between the defnte ntegral and the area under a curve. We approxmate the area under curves by breakng that area up nto a number of peces wth bases on the xaxs and sdes parallel to the yaxs. Dependng on the tops chosen, the peces become ether rectangles, trapezods, or regons whose tops are made of portons of parabolasreferred to respectvely as Upper and Lower Rectangles; the Trapezod Rule; and Smpson's Rule. Crtcal Thnkng Questons 1. What are the upper and lower sums for the regon above the xaxs a. On the nterval [2,5]? to plot). (Ula_.bat) Choose the functon y = x + 2, on [2, 5], wth 20 rectangles (choose 300 ponts b. On the nterval [0,5]? to plot). (Ulb_.bat) Choose the functon y = x + 2, on [0, 5], wth 20 rectangles (choose 300 ponts 2. Use area formulas wth whch you are famlar to fnd the area under y = x + 2, above the xaxs, on the nterval a. [2, 5] b. [0, 5] 3. Whch sum n Q.1ab was correct, the upper or lower?
2 4. a. How could you use both the upper and lower sums to get a better approxmaton to the area under a gven functon? b. What happens to the approxmaton usng the upper or lower sum f you ncrease the number of rectangles? 5. Estmate the area under the curve usng the method you found n Q.4a. Choose the functon y = x , on [0, 5], wth 20 rectangles (choose 300 ponts to plot). You wll now approxmate the area under a curve when the regons are trapezods. 6. Dsplay the graph for the functon y = x , on [0, 5], wth 5 ntervals. (Trapa_.bat) a. What s the trapezodal approxmaton to the area under the curve? b. If you ncrease the number of ntervals selected, what do you thnk wll happen to the error? c. What are the approxmatons to the area wth 10 and wth 20 ntervals? How s the answer wth 20 ntervals related to the answer to Q.5? Explan why ths s so. Dsplay the graph for the functon y = x , on [0, 5], wth 10 ntervals. (Trapb_.bat) Dsplay the graph for the functon y = x , on [0, 5], wth 20 ntervals. (Trapc_.bat) d. How many ntervals does t take to make the error less than 5? Dsplay the graph and choose the functon and nterval necessary to answer the queston above.(trapd.bat) e. If you change the functon to y = x 6, y = x 7 and then y = x 8 on the same regon, why do you thnk the percent error becomes ncreasngly larger than t was for y = x ,on [0, 5]. (Consder whether ths s due to the regons havng larger area or whether t s due to the degree of the functon changng.) 7. Explan why when fve ntervals are selected wth the trapezod rule, the error s larger than when a larger number of ntervals are selected. 8. Suppose the tops of the regons approxmatng the area are parabolc. a. How do you expect the error to compare wth the error usng the prevous methods?
3 The lnked mage cannot be dsplayed. The fle may have been moved, renamed, or deleted. Verfy that the lnk ponts to the correct fle and locaton. b. Why do you thnk ths s so? You wll now work wth Smpson's rule, whch approxmates the area under a curve wth regons whose tops are parabolc. The curve s shown on the graph n red, and the approxmatng regons wth parabolc tops are shown n yellow. Choose y = x , 0 x 5 wth the number of ntervals equal to 6. In ths case [0, 5] s n sx peces dvded by the seven xvalues x = ,,,,,,and 5 The number of parabolas drawn s the number of ntervals dvded by 2. The frst parabola s drawn through the y values correspondng to the frst three xvalues, and the second parabola s drawn through the yvalues correspondng to the thrd through the ffth xvalue, and the thrd parabola s drawn through the yvalues correspondng to the ffth through the seventh xvalues. So, f n equals the number of ntervals, there wll be parabolc tops and they wll touch each other at n 21 places. Thus, on the nterval [0,5], when n = 6, there are parabolas drawn, and they touch at the xvalues. Your answers to questons 9 & 10 below, should be accurate to the fourth place to the rght of the decmal pont. 9. a. The area when n = 6 s, and the error s. (Smpa.bat, n = 6) b. The area when n = 10 s, and the error s. (Smpa.bat, n = 10) c. The area when n = 20 s, and the error s. (Smpa.bat, n = 20) d. The area when n = 60 s, and the error s. (Smpa.bat, n = 60) 10. Choose the functon y = x , 0 x 5. a. The area when n = 6 s, and the error s. (Smpb.bat, n = 6) b. The area when n = 10 s, and the error s. (Smpb.bat, n = 10) c. The area when n = 20 s, and the error s. (Smpb.bat, n = 20) d. The area when n = 60 s, and the error s. (Smpb.bat, n = 60)
4 11. Is the fact that the percent error for Q.10 s larger than the percent error for Q.9 because the regons have larger area, or s t because of some other reason? Explan. 12. Use your text to fnd formulas to predct the maxmum error usng the Trapezod Rule or Smpson's Rule. For each stuaton n Qs. 6, 9 and 10, determne the maxmum predcted error. Show your work. Max. predcted error Q.6 a. c. c.. Max. predcted error Q.9 a. b. c. d. Max. predcted error Q.10 a. b. c. d. Are your computer estmates less than the maxmum predcted error? 13. Rank the approxmaton methods Upper and Lower Rectangles, Trapezod Rule and Smpson's Rule n the order of ther accuracy. You wll now work wth the Remann Sum whch, whle t s not useful as an approxmaton method, wll be used to defne the defnte ntegral later n ths module. Choose the functon y = x , 0 x 5 wth 300 ntervals and wth 20, 50, and then 100 rectangles. (Rem_.bat) (Repeat each selecton at least several tmes. Note that ths method s not useful for ts accuracy, but s used to set up a proof nterrelatng the sums of areas of regons and the defnte ntegral.) 14. a. Do the wdths of the rectangles n each case follow a regular pattern? b. Does each rectangle ht the curve n the same place, e.g., left, mddle, rght, etc.? c. How s ths method of approxmatng the area under the curve dfferent than the method of Upper / Lower rectangles? d. Explan how the area under a curve s estmated usng a Remann Sum. 15. What do you thnk would happen to the error usng the Upper/Lower rectangles, Trapezod, Smpson, or Remann approxmaton, f the number of ntervals chosen became larger and larger? 16. As the number of ntervals becomes larger and larger (as n approaches nf), the sze of each nterval becomes. In the case of the Remann approxmaton, the sze (on average) of the nterval becomes when
5 the number of ntervals ncreases. 17. Let Δx = the wdth of the nterval [x  1, x ], that s, the wdth of one of the Remann rectangles. a. As the x change, the Δx, or we may say the ntervals are wde. b. The (*) n f (x * ) denotes that the xvalue s placed between x  1 and x. c. f (x * ) represents the of the approxmatng regon on the nterval [x  1, x ]. d. f (x * ) Δx represents the of the regon above the nterval [x  1, x ]. e. The symbol f( x) wth =. f. The symbols n * Δx represents the of the products f (x * ) Δx startng wth = and endng = 1 lm max Δx 0 means that the wdth of the nterval goes to. g. Among the contrbutons of F. Remann was hs choce of a very general type of sum so that * lm f( x) max Δx 0 n = 1 Δx s easly calculatedwthout usng lmtsfor certan (ntegrable) functons; and s equal to the under the curve f (x), above the xaxs, between x = a and x = b, when f (x) We now ntroduce the new symbol [a,b], and defne t so that (for the partton of [a, b] as n Q.17g) Fll n the blanks below. b a f(x)dx. called the defnte ntegral of the functon f (x) on the nterval b f(x)dx a * lm f( x) max Δx 0 n = 1 Δx that s, the ntegral of the functon y = f (x) between the xvalues a and b the lmt of the Remann Sum as the mesh of the partton (the length of the largest nterval) goes to zero. In Part II of ths module, you wll use b a f(x)dx to calculate areas of regons, usng a very easly appled theorem. 19. Suppose f (x) s below the xaxs between x = a and x = b. a. What do you thnk wll happen to the area estmate usng one of the approxmatng rules?
6 Hnt: Try Smpson's rule on the computer. (Smp_.bat) b. Why does ths happen? 20. Consder the regon between the xaxs and the functon y =  x on the nterval [0, 5] a. What do you predct the computer approxmaton to the area wll be? b. Verfy your answer to Q.20a usng one of one the methods Upper / Lower Rectangles, Trapezod Rule, or Smpson's Rule. c. Explan your answer. d. What s the area? The Mean Value Theorem In ths module you wll nvestgate whether a certan property always holds true for contnuous functons on a gven nterval. By analyzng a varety of dsplayed functons exemplfyng ths property, you wll determne when ths resultcalled the Mean Value Theorems vald. Use and accept the default left and rght endponts, the default values of a and b and the default functon f (x). The graph wll draw the gven functon wth a secant lne drawn n blue through the ponts (a, f (a)) and (b,f (b)). Crtcal Thnkng Questons 1. How s the slope of the red lne that touches the graph of y = x 33x 2 related to the slope of the blue secant lne? 2. The red lne s to the graph of the functon. 3. What are the coordnates of the two ponts through whch the blue secant lne passes? 4. What s the slope of the secant lne?
7 5. Snce the red lne s to the graph of f (x), we could determne ts slope by fndng the of the functon f (x) at the value of the pont where the red lne hts the graph. 6. Therefore, the slope of the secant lne s to the slope of the to the graph of f (x) at some value between a and b. (Ths result s called the Mean Value Theorem.) 7. a. For the above result to be true, must a =  b? b. Explan n words why (or why not) ths result always holds true for contnuous functons on some nterval [a,b]? (Draw pcture(s) to justfy your concluson.) Skll Exercses 1. For the gven values of a and b and the gven functon n the CTQ above, set up and solve an equaton to determne the xvalue where the red lne must ntersect the graph for t to be parallel to the blue lne. (Hnt: See CTQ 6. above.) 2. Repeat the above exercse for a contnuous functon of your choosng. 3. State the Mean Value Theorem for a functon f(x), wth secant lne through the ponts (a, f (a)) and (b, f (b)). Part II An Applcaton of the Integral to Rocket Propulson The second stage of a twostage rocket s gnted at tme, t = 0, and travels vertcally upward. When the secondstage rocket s 2000 meters above the ground, ts velocty s 250 meters per second (m/s). The rocket at ths tme s programmed to undergo a nonconstant acceleraton, a = 12 t m/s 2, for a burntme of 5 seconds (s). The followng exercses wll drect you toward fndng the maxmum velocty and subsequent maxmum alttude above ground level, h, whch the rocket obtans. 1. You wll frst determne the equaton for the rocket's acceleraton, a (t). a. What s a (t) for t < 5? b. Is there acceleraton provded by the rocket engne after t = 5s? c. What causes acceleraton after t = 5s?
8 d. What s a (t) for t > 5s? e. Sketch a graph of a (t) on the grd below. 2. You wll now determne the equaton for the velocty of the rocket, v (t). a. v (t) s the of a (t). b. v (t) =. Determne ths for both t <= 5 and for t > 5. c. The area under a (t), 0 <= t <= 5, equals the n v (t), 0 <= t <= 5. Ths value s. d. Graph v (t) on the grd below and use the values from Q.2c as part of the graph's label. 3. You wll now determne the equaton for the heght of the rocket, h (t).
9 a. h (t) s the of. b. h (t) =. Determne ths for both t <= 5 and for t > 5. c. The area under v (t), 0 <= t <= 5, equals the n h (t), 0 <= t <= 5. Ths value s. 4. You wll now determne the maxmum alttude. a. What s the maxmum heght obtaned by the rocket? b. What s the velocty at the maxmum alttude? c. What s the acceleraton at the maxmum alttude? d. The area under v (t) for t > 0 equals the n h (t) for t > 0. e. Graph h(t) on the grd below and use the value from Q.4d as part of the graph's label. Bonus queston for ths module. You are travellng on the nterstate n your car at 60 mph when you see a car comng up on you n your rearvew mrror. Suppose you know that t s travellng at 70 mph and that your car s capable of a constant acceleraton of 5 feet per s 2. You hate people passng you, so you stomp on your accelerator as soon as you see the car ganng on you. Assume that you accelerate at your maxmum capablty the whole tme.
10 1. How far behnd you must the other car be when you start acceleratng so that t wll just come even wth you, but not pass you? 2. How long wll you be acceleratng? 3. How fast wll you be travelng when the other car s even wth you? Explorng Calculus
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