The Definite Riemann Integral
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1 These otes closely follow the presettio of the mteril give i Jmes Stewrt s textook Clculus, Cocepts d Cotexts (d editio). These otes re iteded primrily for i-clss presettio d should ot e regrded s sustitute for thoroughly redig the textook itself d workig through the exercises therei. The Defiite Riem Itegrl Suppose tht f is cotiuous fuctio defied o the itervl,. For y give positive iteger, we defie x x x 1 x x x x 3 3x x x R fx ix i1 R is clled the th right edpoit Riem Sum of f o the itervl,. The defiite Riem itegrl (or just defiite itegrl) of f over the itervl, is deoted y fxdx d is defied to e lim R. Thus, fxdx lim R. 1
2 Exmple Fid the vlue of x 3 dx. 1 Solutio The fuctio we re workig with (clled the itegrd)isfx x 3 d the itervl we re workig with is, 1,. For y positive iteger, we hve x 1 x 1 x x 1 3 x x 1 3 d the th right edpoit Riem sum of f over, is
3 R fx ix Sice i1 i1 3 i1 3 i1 3 i1 3 i1 3 3 x i 3 3 x i 3 i i i 7i 1 9 i i 7i3 3 i i i 7 3 i1 3i i 3 i we see tht x 3 dx 1 lim 1,. lim R lim I coclusio, 3
4 x 3 dx
5 Some Bsic Properties of the Defiite Itegrl 1. If the fuctio f is cotiuous o the itervl,, the fxdx exists.. If fx for ll x,, the fxdx is the re of the regio ouded y the curve y fx, the x xis, the lie x, d the lie x. 3. If fx for ll x,, the fxdx is the egtive of the re of the regio ouded y the curve y fx, the x xis, the lie x, d the lie x. 5
6 4. If fx o some prts of the itervl, d fx o other prts of the itervl,, the fxdx is the siged re determied y the curve y fx, the x xis, the lie x, d the lie y. This mes tht prts of the re of the prts tht lie ove the x xis give positive cotriutio to the itegrl, d the re of the prts tht lie elow the x xis give egtive cotriutio to the itegrl. 6
7 5. The uits of re fxdx uits of fx uits of x. 7
8 Exmple The regio ouded y the x xis d the curve y 1 x over the itervl 1,1 is the upper hlf of the uit circle Without doig y computtios, fid the vlue of x dx. 8
9 Exmple Suppose tht the velocity fuctio of oject movig i stright pth is v 3t 48 where t is mesured i secods d v is mesured i feet/secod. Wht re the uits of 5 vdt d wht is the meig of this defiite itegrl (with respect to the motio of the oject)? The grph of v 3t 48 o the itervl,5 is show elow. Use this grph (d some sic geometry) to fid the vlue of the ove defiite itegrl. 4-1 t v -8-1 grph of v 3t 48 9
10 Extedig the Defiitio of the Defiite Itegrl The defiitio tht we hve give for fxdx ssumes tht. We would like to exted our defiitio to iclude the cses where d. We do this s follows: For y cotiuous fuctio f, we defie fxdx d if, the we defie fxdx fxdx. Hvig mde these ew defiitios, we ow stte some importt properties of the defiite itegrl. 1. If f d g re cotiuous fuctios defied o itervl cotiig the poits d (o mtter wht order these poits re i), the fx gxdx fxdx gxdx.. If f is cotiuous fuctio defied o itervl cotiig the poits d (o mtter wht order these poits re i) d K is costt, the Kfxdx K fxdx. 3. If f is cotiuous fuctio defied o the itervl, d fx for ll x,, the fxdx. 4. If f is cotiuous fuctio defied o the itervl,, d m d M re costts such tht m fx M for ll x,, the m fxdx M. 5. If f is cotiuous fuctio defied o itervl cotiig the poits,, d c (o mtter wht order these poits re i), the fxdx c fxdx c fxdx. 1
11 Exmple Give tht d tht fid the vlue of x dx x dx, 8x 3 1 x dx. 11
12 Exmple The grph of fx x 3 6x 8x o the itervl,5 is show elow. Fid the solute mximum vlue, m, d the solute miimum vlue, M, of f o the itervl,5 d m d M to oti lower d upper estimte of the vlue of 5 x 3 6x 8xdx x 1
13 Exmple Referrig to the grph of fx x 3 6x 8x show elow, estimte which of the itegrls i prts e re positive, which re egtive, d which re zero x 1. x 3 6x 8xdx. 4 x 3 6x 8xdx 3. 5 x 3 6x 8xdx x 3 6x 8xdx x 3 6x 8xdx 13
14 Exmple Give tht d tht fid x 3 6x 8xdx 4 5 x 3 6x 8xdx 5 4, 5 x 3 6x 8xdx. 14
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