Mth 06 Clculus Sec. 5.: The Defiite Itegrl I. Riem Sums A. Def : Give y=f(x):. Let f e defied o closed itervl[,].. Prtitio [,] ito suitervls[x (i-),x i ] of legth Δx i = x i -x (i-). Let P deote the prtitio =x 0 < x < x < < x (-) < x =.. Let P e the legth of the logest suitervl. ( P - orm of the prtitio P.) 4. Choose umer x i i ech suitervl ( xi my e " [ ( )] (!x i ) { } edpt). 5. Form the Riem Sum: S P = f x i NOTE:. f does ot hve to e cotiuous or oegtive o [,]. Therefore, S P does ot ecessrily represet pproximtio to the re uder grph.. x i eed ot e the sme i ech itervl.. Δx i eed ot e the sme legth. 4. Riem Sums re used to pproximte give qutity. 5. To icrese the ccurcy of the sum, decrese the suitervl legth; hece, icrese the umer of suitervls. 6. As the ccurcy of the sum icreses, Riem Sum --> Defiite Itegrl. B. Exmples. Fiite Sums fiite sums re exmple of Riem Sums i which ech suitervl hs the sme legth d the sme x i is chose for ech suitervl.. Give y=x o [,]. Prtitio [,] ito the 4 suitervls: [,.],[.,.5],[.5,.6]&[.6,]. Let x =, x =.4, x =.6, x4 =.9. Fid the Riem Sum usig this iformtio.. Repet usig 4 equl suitervls d x i eig the midpoit of ech suitervl.
II. The Defiite Itegrl A. Def : If f is cotiuous fuctio defied for <x<, we divide the itervl [,] ito suitervls of equl width!x = ". We let x 0 (=), x, x,, x (=) e the edpoits of these suitervls d we choose smple poits x, x,, x i these suitervls, so x i lies i the i th suitervl [xi-, x ]. i The the defiite itegrl of f from to is! f ( x ) = lim f x i " # [ ( )] ( $x) { } %. NOT E:. The defiite itegrl! f ( x ) =! f ( t) dt =! f ( r) dr! f ( x ) is umer; it does ot deped o x.. Becuse we hve ssumed tht f is cotiuous, it c e proved tht the limit i the defiitio lwys exists d gives the sme vlue o mtter how the smple poits x i hve ee chose.. gives the siged re of regio etwee the curve y=f(x) d the x- xis o [,]. B. Th m : All cotiuous fs re itegrle, i.e., if f f is cotiuous o [,], the its defiite itegrl over [,] exists. C. Exmples. Express the limit s defiite itegrl o the give itervl ( ). lim 6x i # x i! " $ %x ; [-, ] =. lim x i! " [( ) # 7] 5 $ %x ; [ 4, 7] = $ c. lim # + 4i &! " ( % ' $ 4& % ' =
. Express the defiite itegrls s limit similr to the style i exmple c ove:! f ( x ) = lim f x i " # where!x = " 9! " x + ) 4. (x % ={ [ ( )] ( $x) } i d x i is the right hd edpoit, i.e., xi = + (!x)i. 4! + x). (x D. Evlutig Defiite Itegrls. Approximtig the Vlue of Defiite Itegrls { } # I the previous sectio we used the re of rectgles % & " [ f ( x i )] (!x ) $ ' pproximte the re uder curve. We ow kow tht the defiite itegrl gives us the siged re etwee curve d the x-xis. Therefore, we c use this method to pproximte the defiite itegrl. If we use midpoits s the x i vlue i the defiitio of Riem sum, we cll it the Midpoit Rule. Midpoit Rule: " $ f x i ( ) = #x f x i= {[ ( )] #x } to [ ( ) + f ( x ) +... + f ( x )] where!x = " d x i = ( x i! + x i )=midpoit of [ x i!, x i ]
. Exmple Use the Midpoits Rule with =5 to pproximte! x. Evlutig the Exct Vlue of Defiite Itegrl. Usig geometry/re of regio to evlute the exct vlue of defiite itegrl Sometimes the oly wy to evlute defiite itegrl is to use geometry, s i the first exmple. " 5.) 5 - x!5 5.)! ( - x) 0
.) Use the grph of g elow to evlute! g 5 0. Usig the defiitio of the defiite itegrl to evlute the exct vlue of defiite itegrl.) Bckgroud.) Sigm Nottio for Fiite Sums! (.) k = + + +... + (.) Exmples (.) 4 i!( ) = " ( ) = (.) j + j =!.) Sum Formuls for Positive Itegers (.)! = ;! c = c ; where c is costt (.)! i = ( +) (.)! i = ( +) ( + ) 6 ( ) (4.) i! = + $ ' " # % &
c.) Alger Rules for Fiite Sums (.) Sum/Differece Rule: k ± k!( ) =!( k ) ±!( k ) (.) Costt Multiple Rule: c k d.) Exmples 0 " (.) k! k + =!( ) = c! ( k ), where c is costt 0! (.) 4k = k =5. Evlutig defiite itegrl usig the limit defiitio..) I the def of the defiite itegrl, where!x = " x i = + (!x)i.! f ( x ) = lim " # [ f ( x i )] ( $x) { } % ; we will tke x i to e the right hd edpoit, i.e.,
.) Exmple (.) Evlute the defiite itegrl usig the limit defiitio.! ( x + x + ) -
E. Properties of the Defiite Itegrls. Properties. = 0. = -! f x c. c! = c " ( ) ; c = costt f. d. k f ( x) ( )! = k [ ] e. ± g ( x ) = ±! g( x) c f. + =. Exmples c Give tht " f ( x) = 7, " f ( x) = 5, " g( x) = & " g( x) =!8,! evlute the followig:. =!!. " f ( x) =! c. "! 6g( x) =! [ ] d. " f ( x) + 4g ( x ) =! e. 5 "!
F. Compriso Properties of the Itegrl. The Compriso Properties. If f ( x)! 0 o [, ] " f ( x) #! 0.. If f ( x)! g( x) o [, ] " f ( x) #! # g( x). c. Mx-Mi Iequlity: If M & m re the mximum d the miimum vlues of f o. Exmple [,], the m! ( ) " f ( x) # " M! ( ) ( ). Show tht the vlue of! si x cot e 4. 0