ROB EBY Blinn College Mathematics Department


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1 ROB EBY Blinn College Mthemtics Deprtment Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob EbyFll 26 Weknowthtwhengiventhedistncefunction, wecnfindthevelocitytnypointbyfindingthederivtiveorinstntneous rte of chnge t tht point. Now we re going to look t it the other wy round. When given the velocity or rte of chnge, we will be ble to estimte the distnce trveled. We know if the velocity of moving object is v nd the object hs trveled t units of time, the totl distnce trveled is given by: Distnce = velocity time =v t 1. If cr trvels t 6 mph, then fter 2 hrs the totl distnce trveled is: 2. Wht if the velocity is not constnt? 4 mph the first hlf hour, 7 mph the next hour nd 55 mph the lst hlf hour. Wht is the totl distnce trveled? 3. A cr ccelertes from stop to highwy speed. The velocity v(t) of the cr in meters per second, t t seconds fter it strts, is mesured nd given in the following tble. Estimte the distnce the cr hs trveled over the first 6 seconds. 4. t v(t) Lower estimte t = 2 Upper estimte t = 2 5. Now we hve some better dt. Agin, estimte the totl distnce the cr hs trveled over the first 6 seconds. t v(t) Upper estimte t = 1 Uower estimte t = 1 6. The rte of sles (in gmes per week) of new video gme is shown in the tble below. Assuming tht the rte of sles incresed throughout the 2week period, estimte the totl number of gmes sold during this period. Time (weeks) Rte of sles (gmes per week) Lower estimte: Upper estimte: Left Sum Right Sum Lower estimte = 5()+5(585)+5(892)+5(1875) = 16,76 Upper estimte = 5(585)+5(892)+5(235)+5(235) = 3,885 Left Sum = 5()+5(585)+5(892)+5(235) = 19,135 Right Sum = 5(585)+5(892)+5(235)+5(1875) = 2851
2 Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob EbyFll 26 ROB EBY Blinn College Mthemtics Deprtment SIDE NOTE: Rectngles do NOT hve to be ll the sme width, but for most problems if they hve the sme width, it will simplify our clcultions. We first need to determine how mny rectngles we will use; let tht be n. To pproximte the totl re, find the re of ech individul rectngle (A = bh) nd sum the re of the rectngles. The bse of the rectngle is commonly referred to s the width (denoted x = b ) where nd b re the endpoints of n the intervl [, b]. Before we clculte the height of the rectngle, we hve to decide whether we wnt: left rectngles, L n right rectngles, R n Nottion: x = b n when [,b] is the intervl n = # of rectngles. i= refers to box # [x i 1,x i ]= subintervl x i = refers to the representtive point f(x i ) = height (FACT: [x,x 1 ] = [,+ x], x i+1 = x i + x, [x n 1,x n ] = [b x,b]) so the pproximte re is given by the sum: Def: definite integrl where x i A = [f(x 1 )+f(x 2 )+...+f(x n )] x = n f(x i ) x could be either right sum or the left sum. b f(x)dx = lim n i=1 i=1 n f(x i ) x Write the definite integrl to represent the shded re in the grph below. 3 f(x)=2x^ xdx Mens to find the exct re under the curve using geometry f(x) = 2x from x = 1 to x2 dx
3 Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob EbyFll 26 ROB EBY Blinn College Mthemtics Deprtment Is the estimte n over or under estimte? 2 (x 2 +1)dx n = 2 n = 4 left right exct How cn you tell if the re is n over/under estimte? Wht is the exct re? Def. A function is monotone over n intervl [,b] if it is either lwys incresing over [,b] or lwys decresing over [,b]. Summry monotonic left right incresing decresing Clcultor Informtion Using the TI83 progrm RIEMANN This progrm will do the tedious work of constructing the left or righthnd sums for you for ny specified number of rectngles. It will lso compute midpoint sum, where the height of ech rectngle is the function vlue t the middle of ech rectngle. If the number of rectngles is 32 or less, the progrm will lso drw grph showing the rectngles, if desired. To use the progrm to estimte the vlue of the definite integrl Enter f(x) into Y1. Press [PRGM], move the cursor to RIEMANN, nd press [ENTER]. b f(x)dx follow these steps: On the clcultion screen, prgmriemann will pper on the current line. Press [ENTER], nd the progrm will begin running. Enter the number of rectngles nd the vlues for nd b when sked. Choose sum to construct (lefthnd, righthnd, or midpoint). If you hve chosen 32 rectngles or less, you my elect to view the grph. After the grph is drwn, the clcultor will puse. To continue the progrm, press [ENTER]. The clcultor will return the pproximte vlue for the sum. Evluting Definite Integrls The TI83 hs builtin commnd for evluting integrls: the fnint( ) commnd. It is option 9 of the MATH menu. 4 Exmple: To evlute the integrl e x2 dx, on the homescreen enter the commnd: fnint(e ˆ(x 2 ),x,4,4) nd the 4 clcultor returns the nswer Advntge over using RIEMANN: fnint() provides quick wy to get firly ccurte pproximtion to definite integrl. Disdvntges of using fnint(): No grph is vilble; you cn t esily see if the function hs negtive nd/or positive regions.
4 Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob EbyFll 26 ROB EBY Blinn College Mthemtics Deprtment If the grph of f(x) is positive, the re under the curve is b f(x)dx Discussion: Find 2 (x 2 4)dx Find 2 4 x 2 dx 2 2 Wht does the negtive represent? velocity position with trvel out nd bck Exmples: 1. Given the grph of f(x) nswer the following: Given the grph of f(x) nswer the following: f(x) () Wht is the TOTAL re from x = to x = 4? Wht is the TOTAL re from x = 6 to x = 6? (b) Find 4 f(x)dx Find 6 6 f(x)dx 2. Oil is leking out of ruptured tnker t the rte g(t) gllons per minute. Wht re the units of 5 g(t)dt? nd Wht does tht represent? 3. A culture of bcteri contins 5 million bcteri t t =. If the popultion is growing t the rte of r(t) = (1.5) t million bcteri per hr, () wht does 2 r(t)dt represent? (b) Wht does 2 r(t)dt represent? 1 (c) How mny bcteri re there totl fter t = 2 hours?
5 Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob EbyFll 26 Definition: Fundmentl Theorem of Clculus: If f(x) is continuous on [,b] nd F (x) = f(x) then ROB EBY Blinn College Mthemtics Deprtment b f(x)dx b F (x)dx = F(b) F() π/4 sintdt 3 e 2x dx 4 1 x 3 dx Exmples: 4. Grph of f = F nd F() = 1 find F(b) for b = 1..6 F(1) = 6 find F(t) for t = 1,t = 4 F =f (1,3) (4,9)
6 Mthemtics Deprtment ROB EBY Blinn College Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob EbyFll Suppose we hve n re function defined by n integrl. Then (1) The re function if the bse function is continuous. (2) The re function is lso differentible. A(x) = x r f(t)dtforr x b A (x) = d dx x r f(t)dt 6. Find the derivtive of the following functions d dx x 3 (r 2 +r +1)dr d dx 1 x t4 +1dt d dr r3 cost 2 dt 7. Mtch ech of the functions with its re function A B C D
7 ROB EBY Blinn College Mthemtics Deprtment Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob EbyFll Pssengers for Southwest Airlines flight strt lining up t the gte checkin counter to check in long before the flight tkes off. Agents open the counter for check in 9 minutes before the flight is scheduled to deprt. Once the counter opens, the gents cn serve 2 customers per hour. The grph below gives the rrivl rte (in pssengers per hour) for pssengers joining the line (or pproching the counter if there is no line.) The flight is scheduled to deprt t 5: p.m. The gents open the counter for checkin t 3:3 p.m : 2:3 3: 3:3 4: 4:3 ** The horizontl xis is time, the verticl xis is people per hour 9. How mny pssengers re in line t 3:3? 5: 5:3 1. Assuming tht everyone in line eventully gets checked in nd bords the plne, how mny pssengers bord the plne? 11. When is the line the longest? How long is it t tht point? 12. Does the line ever vnish? If so, when? If not, why not? 13. When is the line incresing in length the fstest? 14. Do ll pssengers bord the plne in time?
8 ROB EBY Blinn College Mthemtics Deprtment Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob EbyFll This grph shows the rrivl rte (in crs per hours) of crs rriving on cmpus nd trying to enter the prking structure. The entrnce gtes will llow 2 crs per hour to enter the grge. After noon, the rtes continues s shown : 7:3 8: 9: 1:: 8:3 9:3 1:3 11:3 12: ** The horizontl xis is time, the verticl xis is rrivl rte per hour. 16. How long is the line t 1: AM? 17. At 1: AM one of the ccess mchine breks nd no one cn enter through tht gte. The rte t which crs cn enter the grge drops to 1 crs per hour. How long is the line t 11: m? 18. At 11: m the other ccess mchine breks nd no one cn enter the grge t ll. At 11:3 m both the gtes re repired nd the rte t which crs cn enter the grge rises gin to 2. Wht is the sitution t 12:? Are there still crs witing to enter the grge? If so, how mny? 19. When is the line the longest? How long is it t tht point? 2. Does the line ever vnish? If so, when? If not, why not? 21. When is the line incresing in length the fstest?
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