Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.166.ICN Textbook: Calculus Early Vectors Preliminary Edition By Stewart. Wed, Jan 22 6.3 Definite Integral p386: 1,4,7,8,10,15,18,19 23,24,(28),29,30 33,34,37,39,42,44,45,(46),49,50,(60) Main Idea: We approximate the area under a curve with rectangles. It seems to work. Key Words: Riemann Sum, Mid Point Rule, Limit f(xi*)(pi-pi-1) |P|->0 Goal: Become familiar with the Riemann Sum. Previous Assignment p377: (6),(16),(20) Page 377 Problem 6 You are given a function f, an interval, partition points, and a description of the points xi* within the ith subinterval. (a) Find ||P||. (b) Find the sum of the areas of the approximating rectangles, as given in (1). (c) Sketch the graph of f and the approximating rectangles. 1 f(x) = ------- [0,2] {0, 1/2, 1, 3/2, 2} x+1 xi* = {1/4, 1, 5/4, 2} (a) Mesh = 1/2. 1 1 1 1 (b) 1/2 ( -------- + ---------- + --------- + ----------- ) 5/4 2 9/4 3 = 1/2 ( 4/5 + 1/2 + 4/9 + 1/3) = 1/2 1/90 (72 + 45 + 40 + 30) = 187/180 = 1.03889 f[x_] := 1/(x+1); P = {0,1/2,1,3/2,2}; S = {1/4,1,5/4,2}; a = Plot[f[x],{x,0,2}]; b = Table[ Plot[ f[S[[i]]],{x,P[[i]],P[[i+1]]}],{i,1,4}]; c = Table[ ParametricPlot[{P[[i+1]],y},{y,0, f[ S[[i]] ] }],{i,1,4}]; Show[a,b,c,PlotLabel->"Page 377 Problem 6"]; ------------------------------------------------------------------------------- Page 377 Problem 16 Use (2) to find the area under the given curve from a to b. Use equal subintervals and take xi* to be the right endpoint of the ith subinterval. Sketch the region. y = x^3 + 2x [0,2]. n Area = SUM (2/n)[ (2i/n)^3 + 2(2i/n) ] i=1 n n Area = 16/n^4 SUM i^3 + 8/n^2 SUM i i=1 i=1 n^2 (n+1)^2 n(n+1) Area = 16/n^4 ------------------ + 8/n^2 ------------ 4 2 4(n+1)^2 4(n+1) Area = --------------- + -------------- n^2 n 4n^2 + 8n + 4 4 n + 4 Area = -------------- + ------------------ n^2 n 8n+4 4 Area = 4 + ------------ + 4 + ------- n^2 n Limit(Area) = 8. --------------------------------------------------------------------- Page 377 Problem 20 Use a calculator to find the region under y = 1/x^2 on [1,2] using n = 10,30,50. f[n_] := (1/n) Sum[ 1/(1+i/n)^2,{i,1,n}]; f[10] = 0.463955 f[30] = 0.487662 f[50] = 0.492558 f[1000] = 0.499625 The answer is 1/2. RIEMANN SUM ============ The definition of a Riemann Sum for a function y = f[x] on an interval [a,b]. (1) Choose a partition p < p < p ... < p where p = a and p = b. 0 1 2 n 0 n (2) Choose a sample point x* ,x* ,x* , ..., x* with x* in [p ,p ]. 1 2 3 n i i-1 i (3) The value of the Riemann Sum is n SUM f(x* ) (p -p ). i i i-1 This value corresponds to an approximation of the area under the curve. If the x* are chosen to be at the minimal values of the function, then i the Riemann sum is the Lower Sum. If the x* are chosen to be at the maximal values of the function on the i interval, then the Riemann sum is the Upper Sum. b INT f(x) dx = Limit SUM f(x* )(p -p ) a mesh->0 i i i-1 (a) When f(x) is continuous the limit exists. (b) When f(x) is increasing the limit exists even if f is not continuous. (c) If f(x) goes off to infinity, the limit does not exist. (d) If m <= f(x) <= M for x in [a,b], then m(b-a) <= SUM f(x* )(p -p ) <= M(b-a) i i i-1 (e) In general, we perceive that a < b. For the rest of the cases: a INT f(x) dx = 0 a a b INT f(x) dx = - INT f(x) dx b a This is not unreasonable. Since as the partition now runs from larger to smaller, the lengths of the cells are now negative. In class Problems: (a) Show that for the constant function f(x) = c, the value of the Riemann sum is c(b-a) for any partition and any choice of x* . i (b) Show that any function y = f(x), any partition p < p < p < ... < p , and any choice of x* ,x* , .., x* , that 0 1 2 n 1 2 n Lower Sum <= Riemann Sum <= Upper Sum (c) Show that for an increasing function y = f(x), Upper Sum - Lower Sum <= (f(b)-f(a))* (mesh size) (d) How would you use the Riemann Sum type idea to estimate the number of people in a photograph? (e) How would you use the Riemann Sum type idea to estimate the area of Akpatok Island? (f) Given functions f and g, a partition, a=p < p < p < ... < p =b, 0 1 2 n and a sample x* , x* , ... , x* , then 1 2 n Riemann Sum(f+g) = Riemann Sum(f) + Riemann Sum(g) (g) Given function f, a partition, a=p < p < p < ... < p =b, 0 1 2 n and a sample x* , x* , ... , x* , then for any constant c, 1 2 n Riemann Sum (cf) = c Riemann Sum(f). (h) Show that the Riemann Sum for y = 1/x on [0,1] does not have a limit.