INTRODUCTION TO INTEGRATION


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1 INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide the intervl [, b] into n subintervls of equl length. Denote this equl length by x. Note tht x = STEP 2: In the i th subintervl, where 1 i n, choose smple point nd denote tht smple point by x i. STEP 3: Form n rectngles, one on ech subintervl, whose heights re the function vlues on the smple points. The i th rectngle is on the i th subintervl nd hs: width= length= The re of the i th rectngle is: The sum of the res of ll n rectngles is:
2 Exmple: Estimte the re A under the grph of f(x) = x 2 from x = 1 to x = 3 using four rectngles nd midpoints. Sketch the rectngles. Nottion: Since n = 4 nd x i is the midpoint of the i th subintervl
3 Exmple: Estimte the re A under the grph of f(x) = x from x = 0 to x = 3 using n = 3 nd x i = left endpoint of the i th subintervl. Sketch the rectngles. NOTE:
4 Exmple: Estimte the re A under the grph of f(x) = x from x = 0 to x = 3 using n = 3 nd x i = right endpoint of the i th subintervl. Sketch the rectngles. NOTE:
5 Exmple: Speedometer redings for motorcycle t 12 second intervls re given in the tble below. t(s) v (ft/s) ) Estimte the distnce trveled by the motorcycle during this time period using the velocities t the beginning of the time intervls. b) Give nother estimte using the velocities t the end of the time periods.
6 5.2 The Definite Integrl Sigm nottion is n bbrevited method of writing sums. Exmple: 5 i = Here Σ is the cpitl Greek letter sigm nd i is clled the index of the summtion. In generl: finish strt of index (objects to sum) Exmple: 1. 5 i = 2. 5 m = m= (k 3) = k= ( 1) j 2 j j=0 Summtion Properties () (c) c = nc ( i + b i ) = i + b i (b) (d) c i = c ( i b i ) = i i b i
7 Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b STEP 1: Divide the intervl [, b] into n subintervls of equl length. Denote this equl length by x. Note tht x = b n STEP 2: In the i th subintervl, where 1 i n, choose smple point nd denote tht smple point by x i. STEP 3: Form n rectngles, one on ech subintervl, whose heights re the function vlues on the smple points. The re of the i th rectngle is: f(x i ) x. The sum of the res of ll n rectngles is: f(x 1) x + f(x 2) x + + f(x n 1) x + f(x n) x = STEP 4: Observe tht s n gets lrger nd lrger, the sum f(x i ) x becomes more nd more ccurte s n pproximtion for A. Thus, we rrive t the following definition: Definition:
8 Definition Let f(x) be ny function defined on n intervl [, b]. Divide [, b] into n subintervls of equl length x = b nd let x i be ny smple point in the i th subintervl, 1 i n. n () The sum f(x i ) x is clled. (b) (c) (d) (e) The limit lim n f(x i ) x is denoted by. f(x) dx is clled the definite integrl of f(x) from to b. If the limit exists nd is finite, then f(x) is sid to be integrble on the intervl [, b]. is the integrl sign, f(x) is the integrnd, is the lower limit of integrtion, b is the upper limit of integrtion, nd dx is clled differentil. The process of computing f(x) dx is clled definite integrtion. Observtions: 1. The definite integrl f(x) dx is limit, just s the derivtive f (x) is limit. 2. f(x) dx= f(t) dt= f(u) du. Wht is importnt is the function f nd the intervl [, b]. For this reson x is sometimes clled dummy vrible. 3. The Riemnn sum f(x i ) x is n pproximtion for the limit The bigger the vlue of n, the better the pproximtion. f(x) dx. Theorem If f is integrble on [, b], then where x = b n nd f(x) dx = x i = + i x. lim n f(x i ) x
9 Fct: on [, b]. f(x) dx = lim n f(x i ) x represents re under the grph of f(x) only when f(x) 0 Question: Wht does f(x) dx represent when f(x) 0 on [, b]? Answer: If f(x) 0 on n intervl [, b], the definite integrl f(x) dx represents the net re. Exmple: Evlute the following integrls in terms of res t dt
10 x2 dx (2 x ) dx x 2 dx
11 Question: How do we compute f(x) dx in generl? Answer: 1. For the vst mjority of functions f(x), the definite integrl exctly. So ll we cn do is to pproximte 2. Sometimes f(x)dx by Riemnn sums. f(x) dx cnnot be computed f(x) dx represents the re or the net re of simple geometric objects like tringles or prts of circles nd so cn be computed esily. 3. For smll but importnt fmily of functions f(x), the definite integrl f(x) dx cn be clculted using ntiderivtives nd Prt II of the Fundmentl Theorem of Clculus. We will see this in Section 5.4.
12 Properties of the Definite Integrl Let f(x) nd g(x) be continuous functions on nd intervl [, b]. Let c be constnt nd ssume tht both f(x) dx nd g(x) dx re finite. Then, b f(x) dx = f(x) dx = c dx = [f(x) ± g(x)] dx = cf(x) dx = f(x) dx = c f(x) dx + c f(x) dx, for ny c in [, b] Comprison Properties: 7. If f(x) 0 on [, b] then f(x) dx 8. If f(x) g(x) on [, b] then 9. If m f(x) M on [, b], then
13 Exmple: Suppose 5 2 f(x) dx = 7. Find f(x) dx= f(x) dx = Exmple: Suppose 4 g(x) dx = 2 nd 5 g(x) dx = 6. Wht is g(x) dx? Exmple: Show tht 1 0 (1 + cos x) dx 2
14 5.3 The Fundmentl Theorem of Clculus Suppose f(t) is continuous function on n intervl [, b]. Then x b. f(t) dt is net re. Let f(t) b t Thus, x f(t) dt is function of x nd it mkes sense to sk wht is the derivtive of x f(t) dt. The Fundmentl Theorem of Clculus, Prt I: Let f be continuous function on the intervl [, b] nd let g(x) = x f(t) dt for x b. Then NOTE 1. The bove theorem shows tht, in sense, differentition nd integrtion re inverse processes. 2. In the bove theorem the derivtive of g is f. The choice of vribles is not importnt. So, for exmple, g (x) = f(x), g (t) = f(t), g (y) = f(y), etc.
15 The Fundmentl Theorem of Clculus, Prt II: Let f be continuous function on n intervl [, b]. Then ny ntiderivtive of f on [, b]. f(t) dt = F (b) F (), where F is Nottion F (b) F () = [F (x)] b = F (x)]b = F (x) b Exmple: Evlute the following: x 3 dx x dx 3. π 2 0 cos t dt 4. e 1 1 t dt
16 5. π 4 0 tn 2 x dx (x 1) 2 dx x e 1 dx
17 x 2 dx x dx x dx
18 Cution 1 To pply the FTC, Prt II, f must be continuous on [, b]. Exmple: x 2 dx =? Note: f(x) = 1 is not continuous on [ 1, 1], so the FTC, Prt II does not pply. x2 However, F (x) = is n ntiderivtive of f(x) nd F (1) F ( 1) = Cution 2 The FTC, Prt II, hs very wide nd yet limited use becuse for the vst mjority of the functions f(x), their ntiderivtives F (x) re not simple functions. So, for exmple, we cnnot use the FTC to evlute sin(x 2 ) dx, e x2 dx, 1 + x3 dx, etc.
19 5.4 The Indefinite Integrls nd The Net Chnge Theorem Indefinite Integrls The collection of ll ntiderivtives of f is clled the indefinite integrl of f with respect to x, nd is denoted by f(x) dx The symbol is clled the integrl sign, the function f is the integrnd of the integrl, nd x is the vrible of integrtion. Exmple: Evlute the following indefinite integrls: 1. (x 2 2x + 5) dx 2. e x x 2 dx
20 3. x 2 + x + 1 x dx 4. sec x(sec x + tn x) dx 5. 2 x x 2 x dx
21 The Net Chnge Theorem Let F (x) be function nd suppose F (x) is continuous on [, b]. By the FTC, Prt II, we hve: F (x) dx = F (b) F () Thus, integrl of rte of chnge = net chnge. Exmple: Suppose oil is leking from tnker t rte of (t 2 + 1) m 3 /h t t hours. How much oil leked fter 5 hours? Homework: Section 5.1 # 3, 5, 13, 15. Section 5.2 # 3, 9, 35, 37, 38, 41, 42, 43, 48, 49. Section 5.3 # (odd), 45, 49, 51, 55. Section 5.4 # 27, 53, 59, 63, 64, 71.
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