Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus


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1 Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl Theorem of Clculus Compute simple ntiderivtives nd definite integrls Reding: Textbook reding for Unit #9 : Study Sections 5.4, 5.3, 6.2
2 2 Properties of Definite Integrls Exmple: Sketch the re implicit in the integrl π/3 π/3 cos(x) dx If you were told tht sketched. π/3 0 cos(x) dx = 3, find the size of the re you 2
3 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 3 This exmple highlights n importnt nd intuitive generl property of definite integrls. Additive Intervl Property of Definite Integrls f(x) dx = c f(x) dx+ c f(x) dx Explin this generl property in words nd with digrm.
4 4 A more rrely helpful, but eqully true, corollry of this property is second property: Reversed Intervl Property of Definite Integrls f(x) dx = b f(x) dx Use the integrl π/3 0 cos(x) dx+ 0 π/3 cos(x)dx, nd the erlier intervl property, to illustrte the reversed intervl property.
5 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 5 Give rtionle relted to Riemnn sums for the Reversed Intervl property.
6 6 Even nd Odd Functions These properties cn be helpful especilly when deling with even nd odd functions. Define n even function. Give some exmples nd sketch them.
7 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 7 Define n odd function. Give some exmples nd sketch them.
8 8 Integrl Properties of Even nd Odd Functions Find property of odd functions when you integrte on both sides of x = 0.
9 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 9 Find property of even functions when you integrte on both sides of x = 0.
10 10 Linerity of Definite Integrls Exmple: If f(x) dx = 10, then wht is the vlue of Sketch n re rtionle for this reltion. 5f(x) dx?
11 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 11 Exmple: If f(x) dx = 2, nd g(x) dx = 4 then wht is the vlue of f(x)+g(x) dx? Agin, sketch n re rtionle for this reltion.
12 12 Linerity of Definite Integrls kf(x) dx = k f(x)±g(x) dx = f(x) dx f(x) dx± g(x) dx
13 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 13 Simple Bounds on Definite Integrls Exmple: Sketch grph of f(x) = 5sin(2πx), then use it to mke n re rgument proving the sttement tht sin(2πx) 5 2
14 14 Simple Mximum nd Minimum Vlues for Definite Integrls If function f(x) is continuous nd bounded between y = m nd y = M on the intervl [,b], i.e. m f(x) M on the intervl, then m(b ) f(x) dx M(b ) Note tht the mximum nd minimum vlues we get with the method bove re quite crude. Sometimes you will be sked for much more precise vlues which cn often be just s esy to find. Exmple: Use the grph to find the exct vlue of just rnge, but the single correct re vlue sin(2πx) i.e. not
15 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 15 Reltive Sizes of Definite Integrls Exmple: Two crs strt t the sme time from the sme strting point. For the first second, the first cr moves t velocity v 1 = t, nd the second cr moves t velocity v 2 = t 2. Sketch both velocities over the relevnt intervl.
16 16 Which cr trvels further in the first second? Relte this to definite integrl. Comprison of Definite Integrls If f(x) g(x) on n intervl [,b], then f(x) dx g(x) dx
17 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 17 The Fundmentl Theorem of Clculus Reding: Section 5.3 nd 6.2 We hve now drwn firm reltionship between re clcultions (nd 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()
18 18 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 f(x) dx by finding F(x) which is the ntiderivtive of f(x); i.e. function F(x) for which F (x) = f(x).
19 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 19 In other words, if we cn find n ntiderivtive 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 20 Exmple: Use the Fundmentl Theorem of Clculus to find the re bounded ( by ) the xxis, the line x = 2, nd the grph y = x 2. Use the fct tht d 1 dx 3 x3 = x 2.
21 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 21 Sketch the re interprettion of this result.
22 22 We used the fct tht F(x) = 1 3 x3 is n ntiderivtive 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.
23 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 23 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. d Confirm tht dx F(x) = x2.
24 24 With our extensive prctice with derivtives erlier, we should find it strightforwrd to determine some simple ntiderivtives. Complete the following tble of ntiderivtives. function f(x) ntiderivtive F(x) x 2 x 3 3 +C x n x 2 +3x 2
25 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 25 function f(x) ntiderivtive F(x) cosx sinx x+sinx
26 26 function f(x) ntiderivtive F(x) e x 2 x 1 1 x x 2 1 x
27 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 27 The chief importnce of the Fundmentl Theorem of Clculus (F.T.C.) is tht it enbles us (potentilly t lest) to find vlues of definite integrls more ccurtely nd more simply thn by the method of clculting Riemnn sums. In principle, the F.T.C. gives precise nswer to the integrl, while clculting (finite) Riemnn sum gives you no better thn n pproximtion.
28 28 Exmple: Consider the re of the tringle bounded by y = 4x, x = 0 nd x = 4. Compute the re bsed on sketch, nd then by constructing n integrl nd using ntiderivtives to compute its vlue.
29 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 29 Exmple: Use definite integrl nd ntiderivtives to compute the re under the prbol y = 6x 2 between x = 0 nd x = 5.
30 30 The lst entry in our ntiderivtive tble ws f(x) = 1. It is bit of specil x cse, s we cn see in the following exmple. 1 1 Exmple: Sketch the re implied by the integrl x dx. 3
31 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 31 Exmple: Now use the ntiderivtive nd the Fundmentl Theorem of Clculus to obtin the exct re under f(x) = 1 between x = 3 nd x = 1. x Mke ny necessry dpttions to our erlier ntiderivtive tble.
32
33 32 Antiderivtives nd the Fundmentl Theorem of Clculus The F.T.C. tells us tht if we wnt to evlute f(x)dx ll we need to do is find n ntiderivtive F(x) of f(x) nd then evlute F(b) F(). THEREISACATCH.Whileinsomecsesthisrellyisverycleverndstrightforwrd, in other cses finding the ntiderivtive cn be surprisingly difficult. This week, we will stick with simple ntiderivtives; in lter weeks we will develop techniques to find more complicted ntiderivtives. Some generl remrks t this point will be helpful.
34 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 33 Remrk 1 Becuse of the importnce of finding n ntiderivtive of f(x) when you wnt to clculte f(x) dx, it hs become customry to denote the ntiderivtive itself by the symbol f(x)dx The symbol f(x)dx (with no limits on the integrl) refers to the ntiderivtive(s) of f(x), nd is clled the indefinite integrl of f(x) Note tht the definite integrl is number, but the indefinite integrl is function (relly fmily of functions).
35 34 Remrk 2 Since there re lwys infinitely mny ntiderivtives, ll differing from ech other by constnt, we customrily write the ntiderivtive s fmily of functions, in the form F(x)+C. For exmple, x 2 dx = x3 3 +C Note tht n ntiderivtive is single function, while the indefinite integrl is fmily of functions.
36 Unit 9 Definite Integrl Properties; Fundmentl Theorem of Clculus 35 Remrk 3 Since the lst step in the evlution of the integrl f(x)dx, once the ntiderivtive F(x) is found, is the evlution F(b) F(), it is customry to write F(x) b in plce of F(b) F(), s in 4 x 2 dx = x3 4 3 = Remrk 4 The vrible x in the definite integrl 0 0 f(x)dx is clled the vrible of integrtion. It cn be replced by nother vrible nme without ltering the vlue of the integrl. f(x)dx = f(u)du = f(θ)dθ
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