1 Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1
2 Riemnn Sums - 1 Riemnn Sums Are estimtions like LEF T (n) nd RIGHT (n) re often clled Riemnn sums, fter the mthemticin Bernhrd Riemnn ( ) who formlized mny of the techniques of clculus. The generl form for Riemnn Sum is f(x 1 ) x + f(x 2 ) x f(x n) x n = f(x i ) x where ech x i is some point in the intervl [x i 1, x i ]. For LEF T (n), we choose the left hnd endpoint of the intervl, so x i = x i 1; for RIGHT (n), we choose the right hnd endpoint, so x i = x i. i=1 x b x 1 x 2 x 3... x n 1 0 x n x
3 Riemnn Sums - 2 The common property of ll these pproximtions is tht they involve sum of rectngulr res, with widths ( x), nd heights (f(x i )) There re other Riemnn Sums tht give slightly better estimtes of the re underneth grph, but they often require extr computtion. We will exmine some of these other clcultions little lter.
4 The Definite Integrl - 1 The Definite Integrl We observed tht s we increse the number of rectngles used to pproximte the re under curve, our estimte of the re under the grph becomes more ccurte. This implies tht if we wnt to clculte the exct re, we would wnt to use limit. The re underneth the grph of f(x) between x = nd x = b is n equl to lim LEF T (n) = lim f(x n n i 1 ) x, where x = b n. i=1
5 The Definite Integrl - 2 b b b b x x x x
6 The Definite Integrl - 3 This limit is clled the definite integrl of f(x) from to b, nd is equl to the re under curve whenever f(x) is non-negtive continuous function. The definite integrl is written with some specil nottion.
7 The Definite Integrl - 4 Nottion for the Definite Integrl The definite integrl of f(x) between x = nd x = b is denoted by the symbol f(x) dx We cll nd b the limits of integrtion nd f(x) the integrnd. The dx denotes which vrible we re using; this will become importnt for using some techniques for clculting definite integrls. Note tht this nottion shres the sme common structure with Riemnn sums: sum ( sign) widths (dx), nd heights (f(x))
8 The Definite Integrl - 5 Problem. Write the definite integrl representing the re underneth the grph of f(x) = x + cos x between x = 2 nd x = 4.
9 The Definite Integrl nd LEFT vs RIGHT - 1 The Definite Integrl - LEF T (n) vs RIGHT (n) s n We might be concerned tht we defined the re nd the definite integrl using the left hnd sum. Would we get the sme nswer for the definite integrl if we used the right hnd sum, or ny other Riemnn sum? In fct, the limit using ny Riemnn sum should give us the sme nswer. Let us look t the left nd right hnd sums for the function 2 x on the intervl from x = 1 to x = 3. Problem. Clculte LEF T (2) RIGHT (2) for x dx. Tht is, how big is the difference between these two estimtes of the re under y = 2 x over x = ?
10 The Definite Integrl nd LEFT vs RIGHT - 2 Clculte LEF T (4) RIGHT (4) for x dx.
11 Clculte LEF T (n) RIGHT (n) for 3 1 The Definite Integrl nd LEFT vs RIGHT x dx.
12 The Definite Integrl nd LEFT vs RIGHT - 4 Wht will the limit of this LEF T (n) RIGHT (n) difference be s n? Wht does this tell us bout wht would hppen if we defined the definite integrl in terms of the right hnd sum? f(x) dx = lim LEF T (n) vs. n lim RIGHT (n)? n
13 Negtive Integrl Vlues - 1 Negtive Integrl Vlues So fr we hve only delt with positive functions. Will the definite integrl still be equl to the re underneth the grph if f(x) is lwys negtive? Wht hppens if f(x) crosses the x-xis severl times? Problem. Suppose tht f(t) hs the grph shown below, nd tht A, B, C, D, nd E re the res of the regions shown. If we were to prtition [, b] into smll subintervls nd construct corresponding Riemnn sum, then the first few terms in the Riemnn sum would correspond to the region with re A, the next few to B, etc.
14 Negtive Integrl Vlues - 2 Which of these sets of terms hve positive vlues? Which of these sets hve negtive vlues?
15 Negtive Integrl Vlues - 3 Problem. Express the integrl res A, B, C, D, nd E. () f(t) dt = A + B + C + D + E f(t) dt in terms of the (positive) (b) (c) (d) f(t) dt = A - B + C - D + E f(t) dt = -A + B - C + D - E f(t) dt = -A - B - C - D - E
16 Negtive Integrl Vlues - 4 If f(t) represents velocity, wht do the negtive res in B nd D represent? () The res B nd D represent negtive positions. (b) The res B nd D represent bckwrds motion. (c) The res B nd D represent distnce trvelled bckwrds.
17 Negtive Integrl Vlues - 5 The Role of Riemnn sums (1) Riemnn sums re needed to sy wht we men by n integrl. (2) Riemnn sums enble us to decide which integrl is pproprite in word problem. (3) Riemnn sums cn lso be used to give n pproximte vlue of the integrl.
18 The Fundmentl Theorem of Clculus - Theory - 1 The Fundmentl Theorem of Clculus We hve now drwn firm reltionship between our Riemnn sums nd re clcultions (nd the physicl properties tht cn be tied to n re clcultion on grph). The time hs now come to build method to compute these res in systemtic wy. The Fundmentl Theorem of Clculus If f is continuous on the intervl [, b], nd we define relted function F (x) such tht F (x) = f(x), then f(x) dx = F (b) F ()
19 The Fundmentl Theorem of Clculus - Theory - 2 The fundmentl theorem ties: the re clcultion of definite integrl bck to our erlier slope clcultions from derivtives. However, it chnges the direction in which we tke the derivtive: Given f(x), we find the slope by finding the derivtive of f(x), or f (x). Given f(x), we find the re defined by f(x) dx by finding F (x) which is the nti-derivtive of f(x); i.e. function F (x) for which F (x) = f(x). In other words, if we cn find n nti-derivtive F (x), then clculting the vlue of the definite integrl requires simple evlution of F (x) t two points (F (b) F ()). This lst step is much esier thn computing n re using finite Riemnn sums, nd lso provides n exct vlue of the integrl insted of n estimte.
20 The Fundmentl Theorem of Clculus - Exmple - 1 Problem. Use the Fundmentl Theorem of Clculus to find the re bounded by the x-xis, the line x = 2, nd the grph y = x 2. Use the fct tht d ( ) 1 dx 3 x3 = x 2.
21 The Fundmentl Theorem of Clculus - Exmple - 2 Problem. We used the fct tht F (x) = 1 3 x3 is n nti-derivtive of x 2, so we were ble use the Fundmentl Theorem. Give nother function F (x) which would lso stisfy d dx F (x) = x2. Use the Fundmentl Theorem gin with this new function to find the re implied by 2 0 x 2 dx.
22 The Fundmentl Theorem of Clculus - Exmple - 3 Did the re/definite integrl vlue chnge? Why or why not? Bsed on tht result, give the most generl version of F (x) you cn think of. Confirm tht d dx F (x) = x2.
23 Bsic Anti-Derivtives - Reference - 1 With our extensive prctice with derivtives erlier, we should find it strightforwrd to determine some simple nti-derivtives. Complete the following tble of nti-derivtives. function f(x) nti-derivtive F(x) x 2 x C x n x 2 + 3x 2
24 Bsic Anti-Derivtives - Reference - 2 function f(x) nti-derivtive F(x) cos x sin x x + sin x
25 function f(x) nti-derivtive F(x) Bsic Anti-Derivtives - Reference - 3 e x 2 x 1 1 x x 2 1 x
26 The Specil Cse of 1/x - 1 The Specil Cse of f(x) = 1 x Problem. Sketch the grphs of f(x) = 1 x nd F (x) = ln(x).
27 The Specil Cse of 1/x - 2 A technicl fct bout integrls is tht if f(x) is continuous on its domin, it must hve n nti-derivtive on tht domin. How is tht seemingly violted by the pir f(x) = 1 x nd F (x) = ln(x)?
28 The Specil Cse of 1/x - 3 Show tht F (x) = ln( x ) stisfies F (x) = 1/x for both positive nd negtive x vlues. in the erlier tble if neces- Correct your nti-derivtive entry for 1 x sry.
29 Anti-Derivtive Prctice - 1 Problem. Find the most generl nti-derivtive of f(x) = x3 3x + 5. x
30 Anti-Derivtive Prctice - 2 Problem. Suppose we wnt to clculte the re under one section of the grph of sin x, the prt from 0 to π. f(x) = sin(x) 1 Then we should clculte A. cos(π) 0 π 2 π B. cos(0) cos(π) C. cos(π) cos(0) D. sin(π) sin(0)
31 Problem. Clculte 4 0 x dx. Anti-Derivtive Prctice - 3
32 Anti-Derivtive Prctice - 4 Nottion The expression F (b) F () comes up so often tht there is specil nottion for it. It is written s b F (x) or [F (x)] b Problem. Clculte π/3 π/4 3 sec 2 (θ) dθ.
33 Properties of Integrls - 1 Properties of Integrls Before we go on to refine our skill t clculting integrls, we should first reflect on some bsic properties of integrls tht derive from their origins s limits of more nd more ccurte Riemnn sums.
34 Properties of Integrls - 2 (1) If > b then (2) If = b then (3) (4) (5) (6) c dx = c (b ). f(x) dx = f(x) dx = 0. [ f(x) ± g(x) ] dx = c f(x) dx = c f(x) dx = c f(x) dx. f(x) dx + b f(x) dx ± c f(x) dx. f(x) dx. g(x) dx.
35 Properties of Integrls - 3 Properties 3-6 need proofs; they depend on the nlogous properties for Riemnn Sums. See Section 5.2 in the textbook for creful proofs of ll of these properties, s well s for properties 7-10 ppering lter.
36 Properties of Integrls - 4 Problem. Prove Property 4 below using the definition of the integrl. [ ] b f(x) g(x) dx = f(x) dx g(x) dx
37 Problem. Consider function for which 2 0 f(x) dx = 3, Use Property 6, 4 1 determine the vlue of f(x) dx = 2 nd f(x) dx = 2 1 c 4 0 f(x) dx + f(x) dx = 4. c Properties of Integrls - 5 f(x) dx, to help f(x) dx. A sketch might help you.
38 Properties of Integrls - 6 Other Properties of Integrls (7) If f(x) 0 for x b then (8) If f(x) g(x) for x b then (9) If m f(x) M for x b then (10) m (b ) f(x) dx f(x) dx M (b ). f(x) dx. f(x) dx 0. f(x) dx g(x) dx.
39 Properties of Integrls - 7 The following digrms illustrte the ides of Properties 8 nd 9, (8) If f(x) g(x) for x b then f(x) dx (9) If m f(x) M for x b then m (b ) M (b ). M f f b g(x) dx. f(x) dx g m b In the second digrm, it cn be observed tht the re under f(x) in [, b ] is greter thn the re under m in [, b ] but less thn the re under M. b
40 Properties of Integrls - 8 Problem. Let f(x) be s shown A C E B D b Sketch f(x) nd so demonstrte one instnce of Property 10, f(x) dx f(x) dx.
41 Net Chnge Theorem - 1 Net Chnge Theorem Note tht we crete n nti-derivtive F (x), we re building it such tht f(x) = F (x). This mens tht f gives the rte of chnge of F. Notice tht this observtion ws mde much erlier, when we strted our discussion of integrtion: when n integrl is ssocited with process of ccumultion then the rte of ccumultion is lwys precisely the integrnd.
42 Net Chnge Theorem - 2 Consider F (x) s the quntity we re trcking, so F is its rte of chnge. Another sttement of the Fundmentl Theorem of Clculus Prt 2 would then be F (x) dx = F (b) F (). The integrl of rte of chnge is the totl chnge. The textbook clls this the Net Chnge Theorem.
43 Net Chnge Theorem - 3 Problem. If cr is moving t v(t) = 2t m/s from t = 1 to t = 5, wht does the quntity 5 1 v(t) dt represent? A. The position of the cr t t = 5, strting t t = 1. B. The net chnge in position of the cr between t = 1 nd t = 5. C. The velocity of the cr t t = 5, strting t t = 1. D. The net chnge in velocity of the cr between t = 1 nd t = 5.
44 Net Chnge Theorem - 4 Problem. If h(t) represents the height of child (in cm) t time t (in yers), nd the child is 120 cm tll t ge 10, which of the following would represent the mount the child grew between t = 10 nd t = 18 yers? A h(t) dt B. C h(t) dt h (t) dt D h (t) dt + 120
45 Net Chnge Theorem - 5 Problem. Suppose wter is flowing into/out of tnk t rte given by r(t) = t L/min, where positive rtes indicte flow in. By how much does the wter level in the tnk chnge during the first 45 minutes fter t = 0?
46 Net Chnge Theorem - 6 Wht is n ssumption you would hve to mke bout the initil mount of wter in the tnk for this to mke sense?