( ) dx ; f ( x ) is height and Δx is

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1 Mth : 6.3 Defiite Itegrls from Riem Sums We just sw tht the exct re ouded y cotiuous fuctio f d the x xis o the itervl x, ws give s A = lim A exct RAM, where is the umer of rectgles i the Rectgulr Approximtio Method. If we do t restrict our fuctios to hvig positive heights AND widths, the we my replce the specil cse ide of re with directed or et re or ccumultio. The Rule of Four represettios of the DEFINITE INTEGRAL, f ( x )dx re:. Verl: The ccumultio of f over the itervl,.. Numericl: The fiite et re (positive, egtive or zero), possessig the uits of f multiplied y the uits of x. 3. Grphicl: The directed re ouded etwee the grphs of the fuctio y = f x ( ) d the x - xis y = from x = to x =.. Alyticl or Limit Defiitio: f ( x )dx = lim x i * i = * f ( x i )Δx, where Δx = d is the x-coordite o the i th rectgle for the give RAM, if the limit exists. Clculus Nottio for Derivtives d Defiite Itegrls: Leiiz s ottio for derivtives dy Δy resemles the limit of the quotiet: lim dx Δx Δx. I the limit, delt for differece hs ee replced with d for derivtive. Leiiz ottio llows us to see the derivtive s ifiitely smll DIFFERENCE QUOTIENT ( comitio of limits, SUBTRACTION & DIVISION). Furthermore, it mkes the Chi Rule meigful: dy dx = dy du du (where y is the outer fuctio of u, dx d u is the ier fuctio of x). Likewise, the limit of the sum of products lim Δx f ( x ) Δx resemles f ( x )dx. Agi, due to the LIMIT, Δx hs ee replced with dx. Now sigm for sum is replced y the elogted S for itegrl, the ifiite SUM of PRODUCTS, or SUMMED PRODUCTS (where A rectgle = l w = f x width of the rectgle). ( ) dx ; f ( x ) is height d Δx is Relee Dufrese 3 of 6

2 Mth : 6.3 Defiite Itegrls from Riem Sums If derivtives re the limit of ifiitely smll d itegrls re the limit of ifiitely smll, wht reltioship do you suspect exists etwee derivtives d itegrls? Defiite Itegrl s Net Are The et or directed (positive, egtive or zero) re etwee the curve f x ( ) d the x xis d ouded etwee the verticl lies x = d x = is give y the DEFINITE INTEGRAL A = f ( x )dx. Theorem: The Existece of Defiite Itegrls All cotiuous fuctios re itegrle. I other words, if fuctio is cotiuous o itervl x,, the its defiite itegrl o x, exists. Exercises: For #-7, sketch the fuctio & shde the regio represeted y ech defiite itegrl. Stte if the defiite itegrl ( et re ) is positive, egtive or. π π. sixdx. si xdx 3. si xdx. si xdx π π π π Relee Dufrese 3 of 6

3 Mth : 6.3 Defiite Itegrls from Riem Sums. 3dx 6. 3dx, where < 7. 3dx, where < 8. Sketch the semi-circle y = x. Express the re etwee the semi-circle x, d the x xis s defiite itegrl d evlute it BY HAND. 9. Evlute x dx o the TI. Is this swer differet from your swer i the previous questio? Expli.. ALL cotiuous fuctios d some discotiuous fuctios re itegrle ( le to hve itegrls or fiite ccumultios over fiite domi). ) Show tht x x dx exists. Iclude sketch i your solutio. 3 Wht kid of discotiuous fuctios (defied o closed domis) re ) itegrle? c) ot itegrle? Relee Dufrese 3 3 of 6

4 Mth : 6.3 Defiite Itegrls from Riem Sums Properties of Defiite Itegrls: Let f d g e fuctios such tht the followig itegrls exist. Furthermore, let x, where. The the followig defiitios d properties of defiite itegrls hold.. Alytic/Limit Defiitio of Defiite Itegrl (y Riem Approx Method): * f ( x )dx = lim f ( x i ) Δx i i =. Verl Defiitio of Defiite Itegrl: f ( x )dx represets the ccumultio of fuctio f over the itervl,. 3. Grphicl Defiitio: f ( x )dx = f ( x ) dx represets the et or directed re etwee the curves y = f x ( ) d y = d the verticl lies x = d x =. ( ) dx ( ), y =, x = d x =. f x represets the (oegtive) re of the regio ouded etwee y = f x Specil Cse: If f x f ( x )dx = f x ( ) dx ( ) for x,, the f x ( ) = f ( x ), d so is the (oegtive) re of the ouded regio. Cutio: I geerl, f ( x )dx f ( x ) dx. Rther, f ( x )dx f ( x ) dx.. Reversig Bouds of Itegrtio: f ( x )dx = f ( x )dx. Zero Width: f ( x )dx = Relee Dufrese 3 of 6

5 Mth : 6.3 Defiite Itegrls from Riem Sums 6. Defiite itegrl of the costt fuctio f x ( ) = : dx = 7. Lierity of Itegrls: ) Homogeeity: k dx = k dx = k ) Distriutivity over dd /sut : f ( x )dx ± g ( x )dx = f ( x ) ± g ( x ) dx ( ) 8. Sudivisio of Itervl, : f ( x c )dx = f ( x )dx + f ( x )dx NOTE: Do ot cofuse this with sudivisio of summtio: c m i = i i = i = + i i =m+ 9. If f ( x ) g ( x ) for x, the f x ( )dx g ( x )dx.. Uder- d Over-Estimtes: If f mi f x ( ) f mx for x, the f mi ( ) f x ( )dx f mx ( ).. Averge Fuctio Vlue: The verge vlue of itegrle fuctio f o the itervl, is the umericl vlue tht the f would e IF f were costt. Let s refer to this costt s f ve. Now we hve two fuctios: y = f x ( ) ( ocostt fuctio, i geerl) d y = f ve ( costt fuctio). If f ve is the verge vlue of the fuctio f, the there ccumultios over the itervl, re equivlet. I other words, their defiite itegrls re equl: A red = A lue f ve dx = f ( x )dx f ve ( ) = f ( x )dx f ve = f ( x )dx Relee Dufrese 3 of 6

6 Mth : 6.3 Defiite Itegrls from Riem Sums Exercises: Properties of Defiite Itegrls. Aswer/stte which property pplies.. Give tht x 3 dx =, evlute the followig BY HAND. ) x 3 + ( )dx d) x 3 dx ) x ( 3 )dx e) x c) x 3 dx ( ) 3 dx. If f ( x )dx = m, f ( x )dx = d g ( x )dx = k, evlute: ) f ( x )dx ) f ( x )dx c) f ( x )dx d) f x ( ) + 3g ( x ) dx 3. Determie the oudry vlues for the itegrl sixdx (costts m d M such tht m sixdx M ) without ctully evlutig the itegrl.. True or flse d expli: xdx x dx o x,. Me Vlue Theorem for Defiite Itegrls: If f is cotiuous o,, the there exists t lest oe x-vlue of c, such tht f c ( ) = f ve = f ( x )dx. Note tht f ( c ) = f ve is the me or verge (fuctio) vlue. Do ot cofuse the verge fuctio vlue (defied y itegrls) with the verge rte of chge of fuctio (defied usig the sect slope d, y the Me Vlue Theorem for derivtives - is equl to the derivtive t c, s log s the fuctio stisfies the theorem s criteri).. For f x ( ) = x 6x + 8 defied o the itervl, 3, evlute ( ) = x 6x + 8 over the itervl, 3 ( ) = x 6x + 8 over the itervl, 3 ) the verge rte of chge of f x ) the verge vlue of f x c) Does the Me Vlue Theorem for Itegrls pply? If yes, t wht poit(s) i the itervl does the fuctio ssume its verge vlue? If o, why ot? Relee Dufrese 3 6 of 6..