1 The Definite Integral As Area


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1 1 The Definite Integrl As Are * The Definite Integrl s n Are: When f () is Positive When f () is positive nd < b: Are under grph of f between nd b = f ()d. Emple 1 Find the re under the grph of y = between = nd = 2. Pge 1 of 13
2 Emple 2 Using the following grph of y = f (), find the vlue of 6 1 f ()d y * Reltionship Between Definite Integrl nd Are: When f () is Not Positive When f () is positive for some vlues nd negtive for others, nd < b: f ()d is the sum of the res bove the is, counted positively, nd the res below the is, counted negtively. Pge 2 of 13
3 Emple 3 For ech of the function f () grphed below, decide whether 3 or pproimtely zero. 3 f ()d is positive, negtive f () f () f () f () Emple 4 Use the following grph of y = f () to estimte 5 3 f ()d 2 1 y Pge 3 of 13
4 Emple 5 Given the grph of y = f () in the below. 1 y () Find 3 f ()d. (b) Find 2 f ()d. (c) Find 5 1 f ()d. (d) Find 3 3 f ()d. Emple 6 Use the following tble to estimte the re between f () nd the is on the intervl f () Pge 4 of 13
5 * Are Between Two Curves If g() f () for b, then Are between grphs of f ()nd g()for b = Alterntively, without the condition g() f (), Are between grphs of f ()nd g()for b = ( f () g())d. f () g() d. Emple 7 Use n definite integrl to find the re under y = 5 ln(2) nd bove y = 3 for 3 5. Pge 5 of 13
6 Emple 8 Find the re between y = + 5 nd y = between = nd = 2. Emple 9 Use definite integrl to find the re enclosed by y = nd y = Pge 6 of 13
7 2 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. Emple 1 The grph of derivtive f () is shown in the following figure. 1 y Fill in the tble of vlues for f () given tht f (3) = f () 2 Pge 7 of 13
8 * Mrginl Cost nd Chnge in Totl Cost If C (q) is mrginl cost function nd C() is the fied cost, Cost to increse production from units to b units = C(b) C() = Totl vrible cost to produce b units = C (q)dq C (q)dq Totl cost of producing b units = Fied cost + Totl vrible cost = C() + C (q)dq Emple 2 The totl cost in dollrs to produce q units of product is C(q). Fied costs re $2,. The mrginl cost is C (q) =.5q 2 q () On grph of C (q), illustrte grphiclly the totl vrible cost of producing 15 units. (b) Estimte C(15), the totl cost to produce 15 units. (c) Find the vlue of C (15) nd interpret your nswer in terms of costs of production. (d) Use prts (b) nd (c) to estimte C(151). Pge 8 of 13
9 Emple 3 A mrginl cost function C (q) is given in the following figure. If the fied costs re $1,, estimte: () The totl cost to produce 3 units. (b) The dditionl cost if the compny increses production from 3 units to 4 units. (c) The vlue of C (25). Interpret your nswer in terms of costs of production. $/unit C (q) q (quntity) Pge 9 of 13
10 Emple 4 The mrginl cost C (q) (in dollrs per unit) of producing q units is given in the following tble. q C (q) () If fied cost is $1,, estimte the totl cost of producing 4 units. (b) How much would the totl cost increse if production were incresed one unit, to 41 units? Emple 5 The mrginl cost function of producing q mountin bikes is C (q) = 6.3q + 5. () If the fied cost in producing the bicycle is $2, find the totl cost to produce 3 bicycles. (b) If the bikes re sold for $2 ech, wht is the profit (or loss) on the first 3 bicycles? (c) Find the mrginl profit on the 31st bicycle. Pge 1 of 13
11 3 Interprettions of the Definite Integrl * The Nottion nd Units for the Definite Integrl The unit of mesurement for f ()d is the product of the units for f () nd the units for. 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. Emple 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 = to t = 2. (b) Find the popultion t time t = 2. Emple 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 = corresponds to Jnury 1, 21. Interpret 9 C(t)dt. Emple 3 Interpret 3 v(t)dt, where v(t) is velocity in meters/sec nd t is time in seconds. 1 Pge 11 of 13
12 Emple 4 A cup of coffee t 9 is put into 2 room when t =. The coffees s temperture is chnging t rte of r(t) = 7(.9 t ) per minute, with t in minutes. Estimte the coffee s temperture when t = 1. Emple 5 A mn strts 5 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? Approimtely how fr wy is he then? (d) When is the mn frthest from his home? How fr wy is he then? Pge 12 of 13
13 Emple 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 =. 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? Emple 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, liters of wter in it on April 1, estimte the quntity of wter in the tower on April 3. rte (liters/dy) t (dys) Pge 13 of 13
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