4.1 Sigma Notation and Riemann Sums

Transcription

1 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas together to get the area of the whole regio. Whe you use this approach with may sub-regios, it will be useful to have a otatio for addig may values together: the sigma (Σ) otatio. summatio sigma otatio how to read it k= k k the sum of k squared, from k equals to k equals the sum of divided by k, from k equals to k equals j=0 a + a + a + a + a 6 + a 7 7 = j the sum of to the j-th power, from j equals 0 to j equals a the sum of a sub, from equals to equals 7 The variable (typically i, j, k, m or ) used i the summatio is called the couter or idex variable. The fuctio to the right of the sigma is called the summad, while the umbers below ad above the sigma are called the lower ad upper limits of the summatio. Practice. Write the summatio deoted by each of the followig: (a) k (b) 7 ( ) j j j= (c) (m + ) m=0 I practice, we frequetly use sigma otatio together with the stadard fuctio otatio: x f (x) g(x) h(x) 0 f (k + ) = f ( + ) + f ( + ) + f ( + ) j= = f () + f () + f () Example. Use the table to evaluate f (x j ) = f (x ) + f (x ) + f (x ) + f (x ) f (k) ad [ + f (j )]. j= Solutio. Writig out the sum ad usig the table values: f (k) = f () + f () + f () + f () = =

2 . sigma otatio ad riema sums 0 while: [ + f (j )] = [ + f ( )] + [ + f ( )] + [ + f ( )] j= = [ + f ()] + [ + f ()] + [ + f ()] = [ + ] + [ + ] + [ + ] which adds up to. Practice. Use the values i the precedig margi table to evaluate: (a) g(k) (b) h(j) j= Example. For f (x) = x +, evaluate (c) [g(k) + f (k )] k= f (k). Solutio. Writig out the sum ad usig the fuctio values: f (k) = f (0) + f () + f () + f () = ( ) ( ) ( ) ( ) = which adds up to 8. Practice. For g(x) = x, evaluate g(k) ad g(k + ). The summad eed ot cotai the idex variable explicitly: you ca write a sum from k = to k = of the costat fuctio f (k) = as f (k) or = + + = =. Similarly: 7 = = = 0 k= Because the sigma otatio is simply a otatio for additio, it possesses all of the familiar properties of additio. Summatio Properties: Sum of Costats: Additio: Subtractio: Costat Multiple: C = C + C + C + + C = C (a k + b k ) = (a k b k ) = C a k = C a k a k + a k b k b k Problems 6 ad 7 illustrate that similar patters for sums of products ad quotiets are ot valid.

3 0 the itegral Sums of Areas of Rectagles I Sectio., we will approximate areas uder curves by buildig rectagles as high as the curve, calculatig the area of each rectagle, ad the addig the rectagular areas together. Example. Evaluate the sum of the rectagular areas i the margi figure, the write the sum usig sigma otatio. Solutio. The sum of the rectagular areas is equal to the sum of (base) (height) for each rectagle: () ( ) + () ( ) + () ( ) = 7 60 which we ca rewrite as k= usig sigma otatio. k Practice. Evaluate the sum of the rectagular areas i the margi figure, the write the sum usig sigma otatio. The bases of these rectagles eed ot be equal. For the rectagular areas associated with f (x) = x i the margi figure: rectagle base height area = f () = = 8 = f () = 6 6 = 6 6 = f () = = 0 so the sum of the rectagular areas is = 7. Example. Write the sum of the areas of the rectagles i the margi figure usig sigma otatio. Solutio. The area of each rectagle is (base) (height): rectagle base height area x x 0 f (x ) (x x 0 ) f (x ) x x f (x ) (x x ) f (x ) x x f (x ) (x x ) f (x ) The area of the k-th rectagle is (x k x k ) f (x k ), so we ca express the total area of the three rectagles as (x k x k ) f (x k ). Practice. Write the sum of the areas of the shaded rectagles i the margi figure usig sigma otatio.

4 . sigma otatio ad riema sums 0 Area Uder a Curve: Riema Sums Suppose we wat to calculate the area betwee the graph of a positive fuctio f ad the x-axis o the iterval [a, b] (see below left). Oe method to approximate the area ivolves buildig several rectagles with bases o the x-axis spaig the iterval [a, b] ad with sides that reach up to the graph of f (above right). We the compute the areas of the rectagles ad add them up to get a umber called a Riema sum of f o [a, b]. The area of the regio formed by the rectagles provides a approximatio of the area we wat to compute. Example. Approximate the area show i the margi betwee the graph of f ad the x-axis spaig the iterval [, ] by summig the areas of the rectagles show i the lower margi figure. Solutio. The total area is ()() + ()() = square uits. I order to effectively describe this process, some ew vocabulary is helpful: a partitio of a iterval ad the mesh of a partitio. A partitio P of a closed iterval [a, b] ito subitervals cosists of a set of + poits {x 0 = a, x, x, x,..., x, x = b} listed i icreasig order, so that a = x 0 < x < x < x <... < x < x = b. (A partitio is merely a collectio of poits o the horizotal axis, urelated to the fuctio f i ay way.) The poits of the partitio P divide [a, b] ito subitervals: These itervals are [x 0, x ], [x, x ], [x, x ],..., [x, x ] with legths x = x x 0, x = x x, x = x x,..., x = x x. The poits x k of the partitio P mark the locatios of the vertical lies for the sides of the rectagles, ad the bases of the rectagles have

5 06 the itegral legths x k for k =,,,...,. The mesh or orm of a partitio P is the legth of the logest of the subitervals [x k, x k ] or, equivaletly, the maximum value of x k for k =,,,...,. For example, the set P = {,,.6,., 6} is a partitio of the iterval [, 6] (see margi) that divides the iterval [, 6] ito four subitervals with legths x =, x =.6, x = 0. ad x = 0.9, so the mesh of this partitio is.6, the maximum of the legths of the subitervals. (If the mesh of a partitio is small, the the legth of each oe of the subitervals is the same or smaller.) Practice 6. P = {,.8,.8,., 6., 7, 8} is a partitio of what iterval? How may subitervals does it create? What is the mesh of the partitio? What are the values of x ad x? A fuctio, a partitio ad a poit chose from each subiterval determie a Riema sum. Suppose f is a positive fuctio o the iterval [a, b] (so that f (x) > 0 whe a x b), P = {x 0 = a, x, x, x,..., x, x = b} is a partitio of [a, b], ad c k is a x-value chose from the k-th subiterval [x k, x k ] (so x k c k x k ). The the area of the k-th rectagle is: f (c k ) (x k x k ) = f (c k ) x k Defiitio: A summatio of the form f (c k ) x k is called a Riema sum of f for the partitio P ad the chose poits {c, c,..., c }. This Riema sum is the total of the areas of the rectagular regios ad provides a approximatio of the area betwee the graph of f ad the x-axis o the iterval [a, b]. Example 6. Fid the Riema sum for f (x) = usig the partitio x {,, } ad the values c = ad c = (see margi). Solutio. The two subitervals are [, ] ad [, ], hece x = ad x =. So the Riema sum for this partitio is: f (c k ) x k = f (c ) x + f (c ) x The value of the Riema sum is.7. = f () + f () = + = 7 0 Practice 7. Calculate the Riema sum for f (x) = x {,, } usig the chose values c = ad c =. o the partitio

6 . sigma otatio ad riema sums 07 Practice 8. What is the smallest value a Riema sum for f (x) = x ca have usig the partitio {,, }? (You will eed to choose values for c ad c.) What is the largest value a Riema sum ca have for this fuctio ad partitio? The table below shows the output of a computer program that calculated Riema sums for the fuctio f (x) = with various umbers x of subitervals (deoted ) ad differet ways of choosig the poits c k i each subiterval. mesh c k = left edge = x k c k = radom poit c k = right edge = x k Whe the mesh of the partitio is small (ad the umber of subitervals,, is large), it appears that all of the ways of choosig the c k locatios result i approximately the same value for the Riema sum. For this decreasig fuctio, usig the left edpoit of the subiterval always resulted i a sum that was larger tha the area approximated by the sum. Choosig the right edpoit resulted i a value smaller tha that area. Why? As the mesh gets smaller, all of the Riema Sums seem to be approachig the same value, approximately.609. (As we shall soo see, these values are all approachig l() ) Example 7. Fid the Riema sum for the fuctio f (x) = si(x) o the iterval [0, π] usig the partitio {0, π, π }, π ad the chose poits c = π, c = π ad c = π. [ Solutio. The three subitervals (see margi) are 0, π ] [ π,, π ] ad [ π ], π so x = π, x = π ad x = π. The Riema sum for this partitio is: ( π ) f (c k ) x k = si π = or approximately.8. π + π + ( π ) + si π ( π + si π = ( + )π 8 ) π Practice 9. Fid the Riema sum for the fuctio ad partitio i the previous Example, but this time choose c = 0, c = π ad c = π.

7 08 the itegral Two Special Riema Sums: Lower ad Upper Sums Two particular Riema sums are of special iterest because they represet the extreme possibilities for a give partitio. We eed f to be cotiuous i order to assure that it attais its miimum ad maximum values o ay closed subiterval of the partitio. If f is bouded but ot ecessarily cotiuous we ca geeralize this defiitio by replacig f (m k ) with the greatest lower boud of all f (x) o the iterval ad f (M k ) with the least upper boud of all f (x) o the iterval. Defiitio: If f is a positive, cotiuous fuctio o [a, b] ad P is a partitio of [a, b], let m k be the x-value i the k-th subiterval so that f (m k ) is the miimum value of f o that iterval, ad let M k be the x-value i the k-th subiterval so that f (M k ) is the maximum value of f o that subiterval. The: LS P = US P = f (m k ) x k is the lower sum of f for P f (M k ) x k is the upper sum of f for P Geometrically, a lower sum arises from buildig rectagles uder the graph of f (see first margi figure) ad every lower sum is less tha or equal to the exact area A of the regio bouded by the graph of f ad the x-axis o the iterval [a, b]: LS P A for every partitio P. Likewise, a upper sum arises from buildig rectagles over the graph of f (see secod margi figure) ad every upper sum is greater tha or equal to the exact area A of the regio bouded by the graph of f ad the x-axis o the iterval [a, b]: US P A for every partitio P. Together, the lower ad upper sums provide bouds o the size of the exact area: LS P A US P. For ay c k value i the k-th subiterval, f (m k ) f (c k ) f (M k ), so, for ay choice of the c k values, the Riema sum RS P = satisfies the iequality: f (c k ) x k f (m k ) x k f (c k ) x k f (M k ) x k or, equivaletly, LS P RS P US P. The lower ad upper sums provide bouds o the size of all Riema sums for a give partitio. The exact area A ad every Riema sum RS P for partitio P ad ay choice of poits {c k } both lie betwee the lower sum ad the upper sum for P (see margi). Therefore, if the lower ad upper sums are close together, the the area ad ay Riema sum for P (regardless of how you choose the poits c k ) must also be close together. If we kow that the upper ad lower sums for a partitio P are withi 0.00 uits of each other, the we ca be sure that every Riema sum for partitio P is withi 0.00 uits of the exact area A.

8 . sigma otatio ad riema sums 09 Ufortuately, fidig miimums ad maximums for each subiterval of a partitio ca be a time-cosumig (ad tedious) task, so it is usually ot practical to determie lower ad upper sums for wiggly fuctios. If f is mootoic, however, the it is easy to fid the values for m k ad M k, ad sometimes we ca eve explicitly calculate the limits of the lower ad upper sums. For a mootoic, bouded fuctio we ca guaratee that a Riema sum is withi a certai distace of the exact value of the area it is approximatig. Theorem: If f is a positive, mootoic, bouded fuctio o [a, b] the for ay partitio P ad ay Riema sum for f usig P, RS P A US P LS P f (b) f (a) (mesh of P) Proof. The Riema sum ad the exact area are both betwee the upper ad lower sums, so the distace betwee the Riema sum ad the exact area is o bigger tha the distace betwee the upper ad lower sums. If f is mootoically icreasig, we ca slide the areas represetig the differece of the upper ad lower sums ito a rectagle: Recall from Sectio. that mootoic meas always icreasig or always decreasig o the iterval i questio. I words, this strig of iequalities says that the distace betwee ay Riema sum ad the area beig approximated is o bigger tha the differece betwee the upper ad lower Riema sums for the same partitio, which i tur is o bigger tha the distace betwee the values of the fuctio at the edpoits of the iterval times the mesh of the partitio. See Problem 6 for the mootoically decreasig case. whose height equals f (b) f (a) ad whose base equals the mesh of P. So the total differece of the upper ad lower sums is smaller tha the area of that rectagle, [ f (b) f (a)] (mesh of P).. Problems I Problems 6, rewrite the sigma otatio as a summatio ad perform the idicated additio.. ( + ). = si(πk). k. ( + j) j=. cos(πj) 6. j=0 k

9 0 the itegral I Problems 7, rewrite each summatio usig the sigma otatio. Do ot evaluate the sums I Problems, use this table: k a k b k to verify the equality for these values of a k ad b k. (a k + b k ) = (a k b k ) = a k = a k a k + a k For Problems 6 8, use the values of a k ad b k i the table above to verify the iequality b k b k ( ) ( ) a k b k = a k b k ( ) a k = a k a k b k = a k b k g (j) 6. j= h(k) 8. = f () h() 0. k g(k) h(i) i= 7 g(k) h(k) I 6, write out each summatio ad simplify the result. These are examples of telescopig sums [ k (k ) ]. 8 [ k ] k + [ k + k ] [k (k ) ] [(k + ) k ] [x k x k ] I 7, (a) list the subitervals determied by the partitio P, (b) fid the values of x k, (c) fid the mesh of P ad (d) calculate x k. 7. P = {,,., 6, 7} 8. P = {,.6,,.,,., 6} 9. P = {,, 0,., } 0. P as show below:. P as show below: For 9 0, f (x) = x, g(x) = x ad h(x) = x. Evaluate each sum f (k) 0. f (j). j=0 g(m). m= i=0 f (k) f ( + i) g ( f (k)). P as show below:

10 . sigma otatio ad riema sums. For x k = x k x k, verify that: x k = legth of the iterval [a, b] For 8, sketch a graph of f, draw vertical lies at each poit of the partitio, evaluate each f (c k ) ad sketch the correspodig rectagle, ad calculate ad add up the areas of those rectagles.. f (x) = x +, P = {,,, } (a) c =, c =, c = (b) c =, c =, c =. f (x) = x, P = {0,,., } (a) c = 0, c =, c = (b) c =, c =., c =. 6. f (x) = x, P = {0,,, 0} (a) c =, c =, c = 9 (b) c = 0, c =, c = 7 7. f (x) = si(x), P = { 0, π, π, π} (a) c = 0, c = π, c = π (b) c = π, c = π, c = π 8. f (x) = x, P = {0,, } (a) c = 0, c = (b) c =, c = For 9, sketch the fuctio ad fid the smallest possible value ad the largest possible value for a Riema sum for the give fuctio ad partitio. 9. f (x) = + x (a) P = {,,, } (b) P = {,,,, } (c) P = {,.,,,, } 0. f (x) = 7 x (a) P = {0,, } (b) P = {0,,, } (c) P = {0,.,,.,, }. f (x) = si(x) (a) P = { 0, π, π} (b) P = { 0, π, π, π} (c) P = { 0, π, π, π}. f (x) = x x + (a) P = {0,, } (b) P = {0,,, } (c) P = {0, 0.,,,., }. Suppose LS P = 7.6 ad US P = 7.0 for a positive fuctio f ad a partitio P of [, ]. (a) You ca be certai that every Riema sum for the partitio P is withi what distace of the exact value of the area betwee the graph of f ad the x-axis o the iterval [, ]? (b) What if LS P = 7.7 ad US P = 7.90?. Suppose you divide the iterval [, ] ito 00 equally wide subitervals ad calculate a Riema sum for f (x) = + x by radomly selectig a poit c k i each subiterval. (a) You ca be certai that the value of the Riema sum is withi what distace of the exact value of the area betwee the graph of f ad the x-axis o iterval [, ]? (b) What if you use 00 equally wide subitervals?. If you divide [, ] ito 0 equally wide subitervals ad calculate a Riema sum for f (x) = + x by radomly selectig a poit c k i each subiterval, the you ca be certai that the Riema sum is withi what distace of the exact value of the area betwee f ad the x-axis o the iterval [, ]? 6. If f is mootoic decreasig o [a, b] ad you divide [a, b] ito equally wide subitervals: the you ca be certai that the Riema sum is withi what distace of the exact value of the area betwee f ad the x-axis o the iterval [a, b]?

11 the itegral Summig Powers of Cosecutive Itegers The formulas below are icluded here for your referece. They will ot be used i the followig sectios, except for a hadful of exercises i Sectio.. Formulas for some commoly ecoutered summatios ca be useful for explicitly evaluatig certai special Riema sums. The summatio formula for the first positive itegers is relatively well kow, has several easy but clever proofs, ad eve has a iterestig story behid it ( ) + = ( + ) k = Proof. Let S = ( ) + ( ) +, which we ca also write as S = + ( ) + ( ) Addig these two represetatios of S together: S = ( ) + ( ) + + S = + ( ) + ( ) S = ( + ) + ( + ) + ( + ) + + ( + ) + ( + ) + ( + ) Karl Friedrich Gauss (777 8), a Germa mathematicia sometimes called the price of mathematics. So S = ( + ) S = ( + ), the desired formula. Gauss supposedly discovered this formula for himself at the age of five whe his teacher, plaig to keep the class busy for a while, asked the studets to add up the itegers from to 00. Gauss thought a few miutes, wrote his aswer o his slate, ad tured it i, the sat smugly while his classmates struggled with the problem. 7. Fid the sum of the first 00 positive itegers i two ways: (a) usig Gauss formula, ad (b) usig Gauss method (from the proof). 8. Fid the sum of the first 0 odd itegers. (Each odd iteger ca be writte i the form k for k =,,,....) 9. Fid the sum of the itegers from 0 to 0. Formulas for other iteger powers of the first itegers are also kow: k = k = k = + ( + ) = + + ( + )( + ) = ( ) ( + ) = k = = ( + )( + )( + ) 0

12 . sigma otatio ad riema sums I Problems 60 6, use the properties of summatio ad the formulas for powers give above to evaluate each sum ( + k + k ) 6. 0 ( ) k k k (k ). Practice Aswers. (a) (b) (c). (a) (b) (c) k = ( ) j j = j= (m + ) = m=0 g(k) = g() + g() + g() + g() = + ( ) + + = 7 h(j) = h() + h() + h() + h() = = j= (g(k) + f (k )) = (g() + f ()) + (g() + f ()) + (g() + f ()) k= = ( + ) + ( + ) + ( + 0) = 0. g(k) = g() + g() + g() = + + = g(k + ) = g() + g() + g() = + + =. Rectagular areas = = = j=. f (x 0 ) (x x 0 ) + f (x ) (x x ) + f (x ) (x x ) = or f (x k ) (x k+ x k ) j j= f ( x j ) ( xj x j ) 6. Iterval is [, 8]; six subitervals; mesh =.; x =.8; x = x x =.8.8 =. ( ) ( ) 7. RS = () + () =. ( ) ( ) 8. smallest RS = () + () = 0.9 ( ) largest RS = ()() + () =. 9. RS = (0) ( π ) + () ( π ) + () ( π ).6