The Bsic Properties of the Itegrl Whe we compute the derivtive of complicted fuctio, like x + six, we usully use differetitio rules, like d [f(x)+g(x)] d f(x)+ d g(x), to reduce the computtio dx dx dx to tht of fidig the derivtives of the simple prts, like x d six, of the complicted fuctio. Exctly the sme techique is used i computig itegrls. Here re uch of itegrtio rules. Theorem 1 (Arithmetic of Itegrtio). Let, d A,B,C e rel umers. Let the fuctios f(x) d g(x) e itegrle o itervl tht cotis d. The () () (c) (d) [f(x)+g(x)] dx [f(x) g(x)] dx [Cf(x)] dx C [Af(x)+Bg(x)] dx A + +B Tht is, itegrls deped lierly o the itegrd. (e) dx Proof. The first three formule re ll specil cses of formul (d). For exmple, formul () is just formul (d) with A B 1. So to get the first four formule it s good eough to just prove formul (d), which we ll do y just comprig the defiitios of the left d right hd sides. Let s itroduce the ottio h(x) Af(x)+Bg(x) for the itegrd o the left hd side. The, y Defiitio 4 i the otes Defiitio of the Itegrl, the left hd side is the limit s of h(x i, ) { Af(x i, )+Bg(x i, ) } { Af(x i,) +Bg(x i,) } Af(x i, ) + Bg(x i, ) (y prt () of Theorem i the otes Defiitio of the Itegrl ) c Joel Feldm. 15. All rights reserved. 1 Jury 9, 15
A f(x i,) +B g(x i,) (y prt () of Theorem i the otes Defiitio of the Itegrl ) Sustitutig the defiitios of d ito A + B gives tht the right hd side is exctly the limit s of the lst lie of (1). Geometriclly, formul (e) just sys tht the re of the rectgle with x ruig from to d y ruig from to 1 is, which is ovious. It is lso esy to see formul (e) lgericlly sice, if we use f(x) 1 to deote the itegrd of the left hd side, the, y Defi itio 4 i the otes Defiitio of the Itegrl, dx lim f(x i, ) lim 1 lim ( ) (1) Exmple I Exmple 1 of the otes Defiitio of the Itegrl, we sw tht ex dx e. So [ e x +7 ] dx e x dx+7 dx (y Theorem 1.d with A 1, f e x, B 7, g 1) (e)+7 (1 ) e+6 (y Exmple 1 i the otes Defiitio of the Itegrl d Theorem 1.e) Exmple Theorem 3 (Arithmetic for the Domi of Itegrtio). Let,,c e rel umers. Let the fuctio f(x) e itegrle o itervl tht cotis, d c. The () () (c) c + c c Joel Feldm. 15. All rights reserved. Jury 9, 15
Proof. For ottiol simplicity, let s ssume tht c d f(x) for ll x. The idetities, tht is Are { (x,y) x, y f(x) } d c +, tht is, c Are { (x,y) } { } x, y f(x) Are (x,y) x c, y f(x) +Are { (x,y) c x, y f(x) } re ituitively ovious. See the figures elow. We wo t give forml proof. y y f(x) y y f(x) x c x So we cocetrte o the formul. The midpoit Riem sum pproximtio to with 4 suitervls is { ( f + 1 +f + 3 +f + 5 +f + 7 )} 4 4 4 4 4 f 8 + 1 5 8 +f 8 + 3 3 8 +f 8 + 5 1 8 +f 8 + 7 )} 8 () 4 We re ow goig to write out the midpoit Riem sum pproximtio to with 4 suitervls. Note tht is ow the lower limit o the itegrl d is ow the upper limit o the itegrl. This is likely to cuse cofusio whe we write out the Riem sum, so we ll temporrily reme to A d to B. The midpoit Riem sum pproximtio to B with 4 suitervls is A { ( f A+ 1 B A +f A+ 3 B A +f A+ 5 B A +f A+ 7 B A )} B A 4 4 4 4 4 f 8 A+ 1 5 8 A+ 3 3 8 A+ 5 1 8 A+ 7 )} B A 8 B 4 Nowrecllig tht A d B, we hve tht themidpoit Riem sum pproximtio to with 4 suitervls is f 8 + 1 5 8 +f 8 + 3 3 8 +f 8 + 5 1 8 +f 8 + 7 )} 8 (3) 4 The curly rckets i () d (3) re equl to ech other the terms re just i the reverse order. The fctors multiplyig the curly rckets i () d (3), mely d, re 4 4 egtives of ech other, so (3) (). The sme computtio with suitervls shows tht the midpoit Riem sum pproximtios to d with suitervls re egtives of ech other. Tkig the limit gives. c Joel Feldm. 15. All rights reserved. 3 Jury 9, 15
Exmple 4 We sw, i Exmple 8 of the otes Defiitio of the Itegrl, tht, whe B >, B xdx B d xdx. By Theorem 3, B xdx d B B xdx xdx B B We my comie the three sttemets B xdx B whe B > ito the sigle sttemet xdx B xdx B whe B > xdx for ll rel umers (Whe >, set B d whe <, set B.) Applyig Theorem 3 yet gi, we hve xdx xdx+ xdx xdx xdx Exmple 4 Exmple 5 Recll tht So x { x if x x if x f ( x ) dx f ( x ) dx+ f( x) dx+ f ( x ) dx Exmple 5 c Joel Feldm. 15. All rights reserved. 4 Jury 9, 15
Exmple 6 Here is cocrete exmple of how the method of Exmple 5 is used. We regoig to compute 1 x dx gi. But this time we re goig to use oly the properties of Theorems 1 d 3 d the fcts tht dx x dx (4) Tht dx is prt (e) of Theorem 1. We sw tht x dx i Exmple 4. The purpose of this exmple is to show how the properties of Theorems 1 d 3 c e used to rewrite [ 1 x ] dx i terms of dx d x dx. First we re goig to get rid of the solute vlue sigs. Recllig tht x x wheever x d x x wheever x, we hve, y Theorem 3.c, 1 1 x dx 1 x dx+ 1 x dx 1 1 ( x) dx+ 1 x dx 1 1+x dx+ 1 x dx Now we pply prts () d () of Theorem 1, d the (4). 1 x dx dx+ x dx+ dx x dx [ ()]+ () 1 +[1 ] 1 Exmple 6 c Joel Feldm. 15. All rights reserved. 5 Jury 9, 15