Introduction to Integration
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1 Introduction to Integrtion Definite integrls of piecewise constnt functions A constnt function is function of the form Integrtion is two things t the sme time: A form of summtion. The opposite of differentition. f (x) = k; where k is (fixed) rel number. The grph of constnt function is horizontl line. A piecewise constnt function is function whose grph is mde up of sections of horizontl lines. T-1 T-2
2 + : : : 6 Grph of - x Definite integrls Suppose f (x) is piecewise constnt function defined on» x» b. The definite integrl of f (x) on» x» b is defined to be Definite integrls in generl f (x) Let be function defined» x» on b. We wnt to define the definite integrl f of (x). To do this we will try to f (x) pproximte by piecewise constnt function. = (vlue of f (x) on 1 st section) (length of 1 st section) + (vlue of f (x) on 2 nd section) (length of 2 nd section) Grph of piecewise constnt pproximtion f (x) to Qk Q b T-3 T-4
3 Piecewise constnt pproximtions Let f (x) be function defined on» x» b. We cn pproximte f (x) by piecewise constnt function s follows: Divide the intervl» x» b into n equl sections. 1 x 2 x 3 x n 1 : : : x q 0 x q x n b 8 Define f (x 0 ) on 1 st section Two importnt observtions: 1. n As gets f bigger (x) n becomes better pproximtion f to (x). 2. f Ech (x) n is piecewise constnt function nd so cn be integrted. This leds to sequence of numbers: n : : : f n (x) = ><. f (x 1 ) on 2 nd section f n (x) dx : : : >: (x n 1) f on lst section. T-5 T-6
4 The previous observtions motivte the following definition: Summry To clculte n pproximtion to The definite integrl of f (x) on» x» b is defined to be using piecewise constnt functions: = f n (x) dx: 1. Divide the intervl into n equl sections. n!1 lim Remrk: If n is lrge then f n (x) dx is 2. Define f n (x) ( piecewise constnt function). 3. Integrte f n (x) to get n pproximtion to good estimte for.. T-7 T-8
5 The Fundmentl Theorem of Clculus Anti-derivtives We hve An nti-derivtive of function f (x) is ny function F (x) such tht = F (b) F () where F (x) is function such tht dx F (x) = f (x): d d F (x) = f (x). dx T-9 T-10
6 x n x 1 = 1 x sin(x) + C cos(x) + C Indefinite integrls A function f (x) hs mny nti-derivtives. Stndrd indefinite integrls Fct: Any two nti-derivtives differ by constnt. f (x) The generl form of n nti-derivtive of f (x) is written s n+1 xn+1 + C ln(x) + C 1 (n6= 1) cos(x) : This expression is clled the indefinite integrl of f (x) with respect to x. e x sin(x) e x + C T-11 T-12
7 = F (b) F (): Rules for indefinite integrls: Using the FTC 1. = To clculte the definite integrl 2. f (x) + g(x) dx using the FTC pply the following steps: 1. Find n nti-derivtive F (x) by working = + g(x) dx out. Indefinite integrls lso stisfy the following differentition property: 2. Use the formul (FTC) = F (x) Λ b d = f (x) Nottion: F (x) Λ b dx = F (b) F () T-13 T-14
8 c g(x) dx c 1 1 Infinite limits of integrtion Rules for definite integrls: We define 1. = b!1 lim = : 2. (x) + g(x) dx f An integrl of the form cn be clculted by pplying the following steps: = + 3. If» b» c then 1. Clculte (which will depend on b). = + : b 2. Determine wht hppens s b! 1. T-15 T-16
9 Summry The following re the lerning outcomes for the mteril covered in Chpter 1: Integrte piecewise constnt functions by summing. Approximte function by piecewise constnt function with specified number of sections. Know the definition of definite integrl for generl function. Clculte numericl pproximtions to definite integrls. Know the Fundmentl Theorem of Clculus. Know nti-derivtives x for for n 6= 1, sin(x), n cos(x), nd 1=x. e x Know the term indefinite integrl. Be ble to use the tble of integrls. Be ble to use the properties of indefinite integrls to integrte combintions of functions. Evlute definite integrls using the Fundmentl Theorem of Clculus. T-17 T-18
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