5 Accumulated Change: The Definite Integral


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1 5 Accumulted Chnge: The Definite Integrl 5.1 Distnce nd Accumulted Chnge * How To Mesure Distnce Trveled nd Visulize Distnce on the Velocity Grph Distnce = Velocity Time Exmple 1 Suppose tht you trvel 30 miles/hour for 2 hours, then 40 miles/hour for 1/2 hour, then 20 miles/hour for 4 hours. () Wht is the totl distnce you trveled? (b) Sketch grph of the velocity function for the trip nd represent the totl distnce trveled on this grph. Exmple 2 Suppose cr is moving with incresing velocity nd suppose we mesure the cr s velocity every two seconds, obtining the following tble. Time(sec) Velocity (ft/sec) () Give lower nd upper estimtes for the distnce this cr hs trveled. (b) On sketch of velocity ginst time, represent the lower nd upper estimtes you hve obtined in (). Lecture Notes  chpter 5 Pge 1 of 25
2 Exmple 3 A cr strts moving t time t = 0 nd goes fster nd fster. Its velocity is shown in the following tble. Estimte how fr the cr trvels during the 12 seconds. Time(sec) Velocity (ft/sec) * Visulizing Distnce on the Velocity Grph: Are Under Curve If the velocity is positive, the totl distnce trveled is the re under the velocity curve. Exmple 4 With time t in seconds, the velocity of bicycle, in feet per second, is given by v(t) = 5t. How fr does the bicycle trvel from t = 0 to t = 10? Lecture Notes  chpter 5 Pge 2 of 25
3 Exmple 5 The velocity, v, of n object (in meters/sec) is given in the following grph. Estimte the totl distnce the object trveled between t = 0 nd t = 6. v(m/sec) t(sec) * Approximting Totl Chnge from Rte of Chnge Exmple 6 A city s popultion grows t the rte of 5000 people/yer for 3 yers nd then grows t the rte of 3000 people/yer for the next 4 yers. Wht is the totl chnge in the popultion of the city during this 7yer period? Lecture Notes  chpter 5 Pge 3 of 25
4 Exmple 7 The rte of sles (in gmes per week) of new video gme is shown in the following tble. Assuming tht the rte of sles incresed throughout the 20week period, estimte the totl number of gmes sold during this period. Time(weeks) Rte of sles (gmes/week) Exmple 8 The following figure shows the rte of chnge of fish popultion. Estimte the totl chnge in the popultion during this 12month period. rte(fish per month) time (months) Lecture Notes  chpter 5 Pge 4 of 25
5 Exmple 9 Two crs strt t the sme time nd trvel in the sme direction long stright rod. The following figure 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? v(km/hr) Cr A Cr B t(hr) Exmple 10 Tow crs trvel in the sme direction long stright rod. The following figure show the velocity, v, of ech cr t time t. Cr B Strts 3 hours fter cr A nd cr B reches mximum velocity of 100 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? v (km/hr) Cr A Cr B t (hr) Lecture Notes  chpter 5 Pge 5 of 25
6 5.2 The Definite Integrl * Left nd RightHnd Sums nd Definite Integrls Let f (t) be function tht is continuous for t b. We divide the intervl [, b] into n equl subdivisions, ech of width t, so t = b n. Let t 0, t 1, t 2,, t n be endpoints of the subdivisions. For lefthnd sum, we use the vlues of the function from the left end of the intervl. For righthnd sum, we use the vlues of the function from the right end of the intervl. Actully, we hve Lefthnd sum = Righthnd sum = n 1 i=0 f (t i ) t = f (t 0 ) t + f (t 1 ) t + + f (t n 1 ) t n f (t i ) t = f (t 1 ) t + f (t 2 ) t + + f (t n ) t i=1 The definite integrl of f from to b, written f (t)dt, is the limit of the lefthnd or righthnd sums with n subdivisions of [, b] s n gets rbitrrily lrge. In other words, f (t)dt = lim n (Lefthnd sum) = lim n ( ) n 1 f (t i ) t i=0 nd f (t)dt = lim n (Righthnd sum) = lim n ( n i=1 f (t i ) t ). Ech of these sums is clled Riemnn sum, f is clled the integrnd, nd nd b re clled the limits of integrtion. Lecture Notes  chpter 5 Pge 6 of 25
7 * Evluting Left nd Righthnd sums Exmple 1 Use the expressions for left nd righthnd sums given on the previous pge nd the following tble. t f (t) () If n = 4, wht is t? Wht re t 0, t 1, t 2, t 3, t 4? Wht re f (t 0 ), f (t 1 ), f (t 2 ), f (t 3 ), f (t 4 )? (b) Find the left nd right sums using n = 4. (c) If n = 2, wht is t? Wht re t 0, t 1, t 2? Wht re f (t 0 ), f (t 1 ), f (t 2 )? (d) Find the left nd right sums using n = 2. * Computing Definite Integrl Exmple 2 Compute 3 1 t2 dt nd represent this integrl s n re. Lecture Notes  chpter 5 Pge 7 of 25
8 * Estimting Definite Integrl from Tble or Grph Exmple 3 Vlues for function f (t) re in the following tble. Estimte 30 f (t)dt by constructing 20 left nd righthnd sums with n = 5. t f (t) Exmple 4 Estimte dx using lefthnd sum with n = 3. x + 1 Lecture Notes  chpter 5 Pge 8 of 25
9 Exmple 5 The function f (x) is grphed in the following figure. Estimte 6 f (x)dx by constructing 0 left nd righthnd sums with n = 6. Drw the corresponding rectngles for both left nd righthnd sums y x Lecture Notes  chpter 5 Pge 9 of 25
10 Exmple 6 Given the grph of y = f (t) in the below. Estimte 6 f (t)dt by constructing left nd 0 righthnd sums with t = 2. Drw the corresponding rectngles for both left nd righthnd sums. y t Lecture Notes  chpter 5 Pge 10 of 25
11 5.3 The Definite Integrl As Are * The Definite Integrl s n Are: When f (x) is Positive When f (x) is positive nd < b: Are under grph of f between nd b = f (x)dx. Exmple 1 Find the re under the grph of y = x between x = 0 nd x = 2. Lecture Notes  chpter 5 Pge 11 of 25
12 Exmple 2 Using the following grph of y = f (x), find the vlue of 6 1 f (x)dx y x * Reltionship Between Definite Integrl nd Are: When f (x) is Not Positive When f (x) is positive for some xvlues nd negtive for others, nd < b: f (x)dx is the sum of the res bove the xxis, counted positively, nd the res below the xxis, counted negtively. Lecture Notes  chpter 5 Pge 12 of 25
13 Exmple 3 For ech of the function f (x) grphed below, decide whether 3 or pproximtely zero. 3 f (x)dx is positive, negtive f (x) f (x) 3 3 x 3 3 x f (x) f (x) 3 3 x 3 3 x Exmple 4 Use the following grph of y = f (x) to estimte 5 3 f (x)dx 2 1 y x Lecture Notes  chpter 5 Pge 13 of 25
14 Exmple 5 Given the grph of y = f (x) in the below. 1 y x 1 () Find 0 3 f (x)dx. (b) Find 2 0 f (x)dx. (c) Find 5 1 f (x)dx. (d) Find 3 3 f (x)dx. Exmple 6 Use the following tble to estimte the re between f (x) nd the xxis on the intervl 0 x 20. x f (x) Lecture Notes  chpter 5 Pge 14 of 25
15 * Are Between Two Curves If g(x) f (x) for x b, then Are between grphs of f (x)nd g(x)for x b = ( f (x) g(x))dx. Exmple 7 Use n definite integrl to find the re under y = 5 ln(2x) nd bove y = 3 for 3 x 5. Lecture Notes  chpter 5 Pge 15 of 25
16 Exmple 8 Find the re between y = x + 5 nd y = 2x + 1 between x = 0 nd x = 2. Exmple 9 Use definite integrl to find the re enclosed by y = 2 + 8x 3x 2 nd y = x. Lecture Notes  chpter 5 Pge 16 of 25
17 5.4 Interprettions of the Definite Integrl * The Nottion nd Units for the Definite Integrl The unit of mesurement for f (x)dx is the product of the units for f (x) nd the units for x. If f (t) is rte of chnge of quntity, then the Totl chnge in quntity between t = nd t = b is given by f (t)dt. Exmple 1 A bcteri colony initilly hs popultion of 14 million bcteri. Suppose tht t hours lter the popultion is growing t rte of f (t) = 2 t million bcteri per hour. () Give definite integrl tht represents the totl chnge in the bcteri popultion during the time from t = 0 to t = 2. (b) Find the popultion t time t = 2. Exmple 2 Suppose tht C(t) represents the cost per dy to het your home in dollrs per dy, where t is time mesured in dys nd t = 0 corresponds to Jnury 1, Interpret 90 0 C(t)dt. Exmple 3 Interpret 3 v(t)dt, where v(t) is velocity in meters/sec nd t is time in seconds. 1 Lecture Notes  chpter 5 Pge 17 of 25
18 Exmple 4 A cup of coffee t 90 is put into 20 room when t = 0. The coffees s temperture is chnging t rte of r(t) = 7(0.9 t ) per minute, with t in minutes. Estimte the coffee s temperture when t = 10. Exmple 5 A mn strts 50 miles wy from his home nd tkes trip in his cr. He moves on stright line, nd his home lies on this line. His velocity is given in the following figure nd positive velocities tke him towrd home. velocity(mph) t (hours) () Does the mn turn round? If so, t wht time(s)? (b) When is he going the fstest? How fst is he going then? Towrd his home or wy? (c) When is he closest to his home? Approximtely how fr wy is he then? (d) When is the mn frthest from his home? How fr wy is he then? Lecture Notes  chpter 5 Pge 18 of 25
19 Exmple 6 The rtes of growth of the popultions of two species of plnts re shown in the following figure. Assume tht the popultions of the two species re equl t time t = 0. new plnts per yer species 2 species t (yers) () Which popultion is lrger fter one yer? After two yers? (b) How much does the popultion of species 1 increse during the first two yers? Exmple 7 The following grph shows the rte of chnge of the quntity of wter in wter tower, in liters per dy, during the month of April. If the tower hd 12,000 liters of wter in it on April 1, estimte the quntity of wter in the tower on April 30. rte (liters/dy) t (dys) Lecture Notes  chpter 5 Pge 19 of 25
20 5.5 The Fundmentl Theorem of Clculus * The Fundmentl Theorem of Clculus The Fundmentl Theorem of Clculus If F (t) is continuous for t b, then F (t)dt = F(b) F(). In words: The definite integrl of the derivtive of function gives the totl chnge in the function. Exmple 1 The grph of derivtive f (x) is shown in the following figure. 1 y x Fill in the tble of vlues for f (x) given tht f (3) = 2. x f (x) 2 Lecture Notes  chpter 5 Pge 20 of 25
21 * Mrginl Cost nd Chnge in Totl Cost If C (q) is mrginl cost function nd C(0) is the fixed cost, Cost to increse production from units to b units = C(b) C() = Totl vrible cost to produce b units = 0 C (q)dq C (q)dq Totl cost of producing b units = Fixed cost + Totl vrible cost = C(0) + 0 C (q)dq Exmple 2 The totl cost in dollrs to produce q units of product is C(q). Fixed costs re $20,000. The mrginl cost is C (q) = 0.005q 2 q () On grph of C (q), illustrte grphiclly the totl vrible cost of producing 150 units. (b) Estimte C(150), the totl cost to produce 150 units. (c) Find the vlue of C (150) nd interpret your nswer in terms of costs of production. (d) Use prts (b) nd (c) to estimte C(151). Lecture Notes  chpter 5 Pge 21 of 25
22 Exmple 3 A mrginl cost function C (q) is given in the following figure. If the fixed costs re $10,000, estimte: () The totl cost to produce 30 units. (b) The dditionl cost if the compny increses production from 30 units to 40 units. (c) The vlue of C (25). Interpret your nswer in terms of costs of production. $/unit C (q) q (quntity) Lecture Notes  chpter 5 Pge 22 of 25
23 Exmple 4 The mrginl cost C (q) (in dollrs per unit) of producing q units is given in the following tble. q C (q) () If fixed cost is $10,000, estimte the totl cost of producing 400 units. (b) How much would the totl cost increse if production were incresed one unit, to 401 units? Exmple 5 The mrginl cost function of producing q mountin bikes is C (q) = q + 5. () If the fixed cost in producing the bicycle is $2000, find the totl cost to produce 30 bicycles. (b) If the bikes re sold for $200 ech, wht is the profit (or loss) on the first 30 bicycles? (c) Find the mrginl profit on the 31st bicycle. Lecture Notes  chpter 5 Pge 23 of 25
24 Focus on Theory * The Second Fundmentl Theorem of Clculus Second Fundmentl Theorem of Clculus If f is continuous function on n intervl, nd if is ny number in tht intervl, then the function G defined on the intervl by G(x) = x f (t)dt hs derivtive f ; tht is, G (x) = f (x). Exmple 1 Suppose tht G(x) = () G (x). (b) G(0). (c) G(7). (d) G (2). x 3 sin t dt. Find t + 6 Exmple 2 Let F(b) = 0 2x dx. () Wht is F(0)? (b) Does the vlue of F increse or decrese s b increses? Assume b 0. (c) Estimte F(1), F(2), F(3). Lecture Notes  chpter 5 Pge 24 of 25
25 Exmple 3 For x =0, 0.5, 1.0, 1.5, nd 2.0, mke tble of vlues for I(x) = x 0 t4 + 1dt. * Properties of the Definite Integrl Sums nd Multiples of Definite Integrls If, b, nd c re ny numbers nd f nd g re continuous functions, then c f (x)dx + c f (x)dx = ( f (x) + g(x))dx = c f (x)dx = c f (x)dx, f (x)dx + f (x)dx. g(x)dx, Exmple 4 Let following integrls. () ( f (x) + g(x))dx (b) (( f (x))2 (g(x)) 2 )dx (c) ( f (x))2 dx ( f (x)dx)2 (d) c f (z)dz f (x)dx = 8, ( f (x))2 dx = 12, g(t)dt = 2, nd (g(t))2 dt = 3. Find the Exmple 5 Given 6 4 f (x)dx = 12, 6 15 g(x)dx = 5, nd 6 integrls. () 6 4 [3 f (x) 5 6 g(x)]dx (b) 4 15 g(x)dx 4 g(x)dx = 18. Find the following Lecture Notes  chpter 5 Pge 25 of 25
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