# Time in Seconds Speed in ft/sec (a) Sketch a possible graph for this function.

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1 4. Are under Curve A cr is trveling so tht its speed is never decresing during 1-second intervl. The speed t vrious moments in time is listed in the tle elow. Time in Seconds Speed in t/sec () Sketch possile grph or this unction. () Estimte the distnce trveled y the cr during the 1 seconds y inding the res o our rectngles drwn t the heights o the let endpoints. This is clled let Riemnn sum. (c) Estimte the distnce trveled y the cr during the 1 seconds y inding the res o our rectngles drwn t the heights o the right endpoints. This is clled right Riemnn sum. (d) Estimte the distnce trveled y the cr during the 1 seconds y inding the res o two rectngles drwn t the heights o the midpoints. This is clled midpoint Riemnn sum.

2 y 1 E. Given the unction, estimte the re ounded y the grph o the curve nd the -is rom = to = y using: () let Riemnn sum with n = 4 equl suintervls () right Riemnn sum with n = 4 equl suintervls (c) midpoint Riemnn sum with n = 4 equl suintervls E. Wter is lowing into tnk over 1-hour period. The rte t which wter is lowing into the tnk t vrious times is mesured, nd the results re given in the tle elow, where is mesured in gllons per hour nd t is mesured in hours. The tnk contins 15 gllons o wter when t =. t (hours) Rt (gl/hr) () Estimte the numer o gllons o wter in the tnk t the end o 1 hours y using let Riemnn sum with three suintervls nd vlues rom the tle. Show the computtions tht led to your nswer. Rt () Estimte the numer o gllons o wter in the tnk t the end o 1 hours y using right Riemnn sum with three suintervls nd vlues rom the tle. Show the computtions tht led to your nswer. Homework: Worksheet

3 4.3 Riemnn Sums nd Deinite Integrls To estimte the re ounded y the grph o nd the -is etween the verticl lines = nd =, prtition the re nd divide it into suintervls. Yesterdy we drew rectngles with the height t the let endpoint or the right endpoint or t the midpoint o the intervl. Tody we will drw rectngles t some generl point within the suintervl, not necessrily t the let endpoint or the right endpoint or t the midpoint o the intervl.. Let c e ny point in the kth suintervl. k Drw rectngle with height o c k. Are o Rectngle = Sum o ll the Rectngles = This sum is clled. How cn we get the ect re under the curve? Are = Deinition o Deinite Integrl: n d lim ckk or d lim ck n k 1 k n k 1 k Are ounded y y nd the -is on [, ] = d

4 Properties o Deinite Integrls: d d k d I c lies etween nd, then d g d g d g d d g d d g nd Note: d g d E. Sketch nd evlute y using geometric ormul. () 6 1 4d 3 () d 1 (c) 4 4 d (d) 4 d nd d 1. Find: E. Given: d () 7 3 () d 7 3 d (c) 5 5 d Homework: P. 77: odd, 3, 3, 46, 47, 49

5 4.1 Antiderivtives nd Indeinite Integrls I you were given 3 nd sked wht unction hd this derivtive, wht would you sy? is clled the o. The symol g d is the. The term is synonym or. We cn get ormuls or ntiderivtives y reversing the dierentition rules: d n n n1 d d d d d d d d d d d d d sin u cos u Dierentition Rules du cos u d du sin u d tn u sec cot u csc du u d du u d du secu secu tn u d du cscu cscu cot u d Integrtion Rules n1 n d C, n 1 n 1 cos u du sin u C sin u du cos u C sec csc u du tn u C u du cot u C secu tn u du secu C cscu cot u du cscu C Properties o Indeinite Integrls g d d g d, where is constnt k d k d k g d d g d nd Note tht E d d g d g d

6 There isn t product rule or quotient rule or ntiderivtives so sometimes you must simpliy irst. 1 3 d E. 7 E. d 6, 1 3 E. Solve the dierentil eqution: cos, 3, E. Solve the dierentil eqution: 3 E. A prticle moves long the -is t velocity o vt 4t 3t 5, t. At time t =. its position is = 3. () Find the ccelertion unction. () Find the position unction. Homework: P. 55: odd, 49, 57, 59, 6-65, 83, 84

7 Discovery o the Fundmentl Theorem o Clculus The other dy we used Geometry to evlute deinite integrls such s 4 1 d 4 1 d. Then we lerned how to ind ntiderivtives. 1 d How re these relted? Let s see i we cn igure it out! Sustitute the lower limit o integrtion, =, nd the upper limit o integrtion, = 4, into your ntiderivtive: I =, I = 4, Wht could we do with the results to get the re tht we ound ove y using Geometry? It turns out tht this lwys works! Steps: 1) Find the ntiderivtive. ) Sustitute the upper limit into the ntiderivtive, then the lower limit into the ntiderivtive, nd then sutrct. This is clled the Fundmentl Theorem o Clculus. Fundmentl ecuse it reltes the two importnt ides o Clculus, derivtives nd ntiderivtives. Now we cn ind the re ounded y grph nd the -is or grphs in which we cnnot use Geometry. Wow! Let s try it on the net prolem. Notice tht you wouldn t e le to ind the vlue y Geometry. E. 1 3 d Now let s write the Fundmentl Theorem symoliclly. Fundmentl Theorem o Clculus: d

8 4.5 Integrtion Using u-sustitution When we dierentited composite unctions, we used the Chin Rule. The reverse process is clled u-sustitution. E. 1 5 d 3 E. 4 5 d E. 3 d E. d 3 4

9 E. cos 3d 3 sin cos d E. 3 sin 5 cos 5 d E. 4 cos sin d E. dy 9 E. Solve the dierentil eqution 5 d 3 1 Homework: P. 36: 17, 1, 5, 7, 33, 37, 39, 41, 47, 49, 53, 55, 59, 61, 63, 64 The ook will slip in some old prolems like the prolems in 4.1 tht don t need u-sustitution so wtch out or those.

10 4.4 Fundmentl Theorem o Clculus d Fundmentl Theorem o Clculus: E. 3 1 d 9 4 E. d 1 E. 4 sec d 3 i E. 1d i

11 E d E. Find the re ounded y the grph o y 3, the -is, nd the verticl lines = nd =. Homework: P. 93: 1 33 odd, odd

12 4.5 u-sustitution with Deinite Integrls E d E. d 1 9 E. 1 sin 3 d

13 E. 6 sin 3 cos d E. Find the re ounded y the grph o y 1 nd the -is on the intervl [, ]. E. Wter is eing pumped into tnk t rte given y is given.. A tle o vlues o Rt t (min.) (gl/min) Rt () Use dt rom the tle nd our suintervls to ind let Riemnn sum to pproimte R t dt. Rt () Use dt rom the tle nd our suintervls to ind right Riemnn sum to pproimte R t dt. Homework: Worksheet on Deinite Integrls nd Are

14 4.5 Another Kind o Sustitution Sometimes the u-sustitution method we hve lerned doesn t work, nd we need to do something dierent to integrte. E. d 5 E. d E. 3 1 d

15 I unction is even, then hs I unction is odd, then hs y-is symmetry so origin symmetry so d d E. Given: Find: () is even nd 5 d 3. E. Given: Find: 5 d = 5 5 = () 5 d is odd nd 5 d 3. () = () 5 d d = = (c) (c) 4 d 5 4 d = 5 Deinition. Let the intervl [, ] e prtitioned into n suintervls y ny L 1 n1 n, n 1 points nd let k k k 1 denote the width o the kth suintervl. Within ech suintervl k, k 1, choose ny smpling point is Riemnn sum with n sudivisions or on [, ]. Deinition. Let unction denoted y S c c L c c c k. The sum n 1 1 n n k k k 1 e deined on the intervl [, ]. The integrl o over [, ], d, is the numer, i one eists, to which ll Riemnn sums to ininity nd s the widths o ll sudivisions tend to zero. In symols, d lim S lim c. n k k n n k 1 E.. I n is positive integer, then epress the ollowing s deinite integrl n lim n n n n n n S n n tend s n tends

16 From old AP Multiple Choice Tests: 1 1 3n 45. I n is positive integer, then lim... cn e epressed s n n n n n (A) 1 1 d 1 1 (B) 3 d (C) 31 d (D) 3 d (E) 3 3 d 1 1 n 41. lim... = n n n n n (A) d (B) 1 d (C) 1 d (D) 1 d (E) d 1 5. The closed intervl [, ] is prtitioned into n equl suintervls, ech o width y the numers (A),,..., where.... Wht is 1 n 1 n (B) 3 (C) i 1, lim n i? n (D) (E) Homework: Worksheet

17 Fundmentl Theorem o Clculus dy Given d with the initil condition y y Method 1: Integrte Find 3. y d, nd use the initil condition to ind C. Then write the prticulr solution, nd use your prticulr solution to ind y 3. Method : Use the Fundmentl Theorem o Clculus: d Sometimes there is no ntiderivtive so we must use Method nd our grphing clcultor. sin nd 1 5. Find. E. Wht i you hd een given nd then were sked to ind 1?

18 E. The grph o consists o two line segments nd 5 semicircle s shown on the right. Given tht ind: (), y () Grph o (c) 6 E. The grph o is shown. Use the igure nd the ct tht () 3 5 to ind: y Are = 4 Are = () (c) 7 9 Are = 9 Then sketch the grph o. y E. A pizz with temperture o 95 C is put into 5 C room when t =. The pizz s.1t temperture is decresing t rte o r t 6e C per minute. Estimte the pizz s temperture when t = 5 minutes. Homework: Worksheet

19 Fundmentl Theorem o Clculus, Dy 8 E. I 3 5, is continuous, nd d, ind the vlue o E. I 3 d 8, ind d , E. 1, Evlute: 4 1 d Homework: Worksheet

20 AVERAGE VALUE OF A CONTINUOUS FUNCTION Men Vlue Theorem or Integrls I is continuous on [, ], then there eists numer c in [, ] such tht d c. The geometric interprettion o the Men Vlue Theorem or Integrls is tht, or positive unction, there is numer c etween nd such tht the rectngle with se [, ] nd height c hs the sme re s the region under the grph o rom to. In other words, c is the vlue o on [, ] where you cn uild perect rectngle--- rectngle whose re is ectly equl to the re o the region under the grph o rom to. c = height o the perect rectngle = se o the perect rectngle Are o perect rectngle = The vlue c is clled the verge vlue o the unction nd is deined y: 1 ve d E. Given 1 nd the intervl [ 1, ], () Find the verge vlue o on the given intervl. () Find c such tht ve c. (c) Sketch the grph o nd rectngle whose re is the sme s the re under the grph o. E. The tle elow gives vlues o continuous unction. Use let Riemnn sum with three suintervls nd vlues rom the tle to estimte the verge vlue o on [5, 17]

21 E. A study suggests tht etween the hours o 1: PM nd 4: PM on norml weekdy, the speed o the tric on certin reewy eit is modeled y the ormul S t t 1t 6t where the speed is mesured in kilometers per hour nd t is the numer o hours pst noon. Compute the verge speed o the tric etween the hours o 1: PM nd 4: PM. (Use your clcultor, nd give your nswer correct to three deciml plces.) 3 E. Suppose tht during typicl winter dy in Minnepolis, the temperture (in degrees Celsius) hours ter midnight is shown in the igure elow. () Use midpoint Riemnn sum with our equl suintervls to pproimte the verge temperture over the time period rom 4: AM to 8 PM. () Use your nswer to () to estimte the time when the verge temperture occurred. y E. Find the verge vlue o the unction on the given intervl without integrting. (Hint: Grph nd use Geometry.) 3 i 1 3 on 1, 6 6 i 3 6 Homework: Worksheet

22 INTEGRATION USING DATA E. Wter is lowing into tnk over 4-hour period. The rte t which wter is lowing into the tnk t vrious times is mesured, nd the results re given in the tle elow, where is mesured in gllons per hour nd t is mesured in hours. The tnk contins 15 gllons o wter when t =. t (hours) (gl/hr) Rt () Estimte the numer o gllons o wter in the tnk t the end o 4 hours y using midpoint Riemnn sum with three suintervls nd vlues rom the tle. Show the computtions tht led to your nswer. Rt () Estimte the numer o gllons o wter in the tnk t the end o 4 hours y using trpezoidl sum with three suintervls nd vlues rom the tle. Show the computtions tht led to your nswer Use the model to ind the 75 numer o gllons o wter in the tnk t the end o 4 hours. (c) A model or this unction is given y W t t t (d) Use the model given in (c) to ind the verge rte o wter low over the 4-hour period. Homework: Worksheet

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