MATH 149LABORATORY ASSIGNMENT 8THE FUNDAMENTAL THEOREMOF CALCULUS In class we defined the definite integral of the function f (x) on the interval [ a, b ] asd⌠⌡⎮⎮ab()f xx = Lim ∑ = k 1n()f ck() − xkx − k 1 where a = x0 < x1 < x2 < ... < x − n 1 < xn = b is a partition of [ a, b ] , x − k 1 < ck < xkfor k = 1...n, and the limit is taken as → n ∞ and the lengths of the subintervals [ x − k 1 , xk ] determined by the partition go to 0. The easiest and best way to evaluate the definite integral is by using the Fundamental Theorem of Calculus, which says thatd⌠⌡⎮⎮ab()fxx = F (b) - F (a)where F is any antiderivative of f ( i.e., F '(x) = f (x) ). In this laboratory we will verify the Fundamental Theorem by first using MAPLE to evaluate the limit of the sum directly and then by finding F and computing F (b) - F (a).EXAMPLE Let f (x) = x cos(πx) , 0 < x < 2π. We will use partitions in which the xk's are equally spaced and each ck is the midpoint of [ x − k 1, xk].First we illustrate the approximating rectangles using a partition with 20 points:> with(student):> f:=x->x*cos(Pi*x); := f → xx()cos π x> middlebox(f(x),x=0..2*Pi,20);The corresponding approximating sum is > middlesum(f(x),x=0..2*Pi,20);110π⎛⎝⎜⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎟∑ = i 019⎛⎝⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟110⎛⎝⎜⎜⎞⎠⎟⎟ + i12π⎛⎝⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟cosπ2⎛⎝⎜⎜⎞⎠⎟⎟ + i1210or> value(%);110π120π⎛⎝⎜⎜⎞⎠⎟⎟cosπ220320π⎛⎝⎜⎜⎞⎠⎟⎟cos3 π22014π⎛⎝⎜⎜⎞⎠⎟⎟cosπ24720π⎛⎝⎜⎜⎞⎠⎟⎟cos7 π220 + + + ⎛⎝⎜⎜920π⎛⎝⎜⎜⎞⎠⎟⎟cos9 π2201120π⎛⎝⎜⎜⎞⎠⎟⎟cos11 π2201320π⎛⎝⎜⎜⎞⎠⎟⎟cos13 π22034π⎛⎝⎜⎜⎞⎠⎟⎟cos3 π24 + + + + 1720π⎛⎝⎜⎜⎞⎠⎟⎟cos17 π2201920π⎛⎝⎜⎜⎞⎠⎟⎟cos19 π2202120π⎛⎝⎜⎜⎞⎠⎟⎟cos21 π2202320π⎛⎝⎜⎜⎞⎠⎟⎟cos23 π220 + + + + 54π⎛⎝⎜⎜⎞⎠⎟⎟cos5 π242720π⎛⎝⎜⎜⎞⎠⎟⎟cos27 π2202920π⎛⎝⎜⎜⎞⎠⎟⎟cos29 π2203120π⎛⎝⎜⎜⎞⎠⎟⎟cos31 π220 + + + +3320π⎛⎝⎜⎜⎞⎠⎟⎟cos33 π22074π⎛⎝⎜⎜⎞⎠⎟⎟cos7 π243720π⎛⎝⎜⎜⎞⎠⎟⎟cos37 π2203920π⎛⎝⎜⎜⎞⎠⎟⎟cos39 π220 + + + + ⎞⎠⎟⎟> evalf(%);1.582743036If we use a partition with n points the approximating sum is> middlesum(f(x),x=0..2*Pi,n);2 π⎛⎝⎜⎜⎜⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎟⎟⎟∑ = i 0 − n 1⎛⎝⎜⎜⎜⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎟⎟⎟2⎛⎝⎜⎜⎞⎠⎟⎟ + i12π⎛⎝⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟cos2 π2⎛⎝⎜⎜⎞⎠⎟⎟ + i12nnnMAPLE knows a formula for this sum:> value(%);2 ππ⎛⎝⎜⎜⎞⎠⎟⎟cosπ2n⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟ − + ⎛⎝⎜⎜⎞⎠⎟⎟sinπ2n21⎛⎝⎜⎜⎞⎠⎟⎟cosπ2n2 − ⎛⎝⎜⎜⎞⎠⎟⎟cosπ2n212⎛⎝⎜⎜⎞⎠⎟⎟sinπ2nπ ()sin π2()cos π2 − ⎛⎝⎜⎜⎞⎠⎟⎟cosπ2n21− − ⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛⎝⎜⎜⎞⎠⎟⎟cosπ2nπ ()cos π22n⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟ − ⎛⎝⎜⎜⎞⎠⎟⎟cosπ2n21π⎛⎝⎜⎜⎞⎠⎟⎟cosπ2n⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟ − + ⎛⎝⎜⎜⎞⎠⎟⎟sinπ2n21⎛⎝⎜⎜⎞⎠⎟⎟cosπ2n2n − ⎛⎝⎜⎜⎞⎠⎟⎟cosπ2n21 − − ⎛⎝⎜⎜⎞⎠⎟⎟cosπ2nπn⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟ − ⎛⎝⎜⎜⎞⎠⎟⎟cosπ2n21 + ⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟n / We now take the limit as → n ∞ ; this will be the exact value of d⌠⌡⎮⎮02 πx ()cos π xx .> limit(%,n=infinity);− + 2()cos π2224π2()sin π2()cos π2π2> evalf(%);1.516185360Next we use MAPLE to find an antiderivative F (x) for f (x). The command is> int(f(x),x); + ()cos π xx()sin π x ππ2(NOTE THAT MAPLE DOES NOT INCLUDE THE " + C".)> F:=x->((cos(Pi*x)+Pi*x*sin(Pi*x))/(Pi^2)); := F → x + ()cos π x π x ()sin π xπ2You should differentiate F (x) "by hand" to check that the derivative is actually f (x)---non-computer methods for finding antiderivatives of functions like f (x) = x cos(πx) will be developed in Math 152.By the Fundamental Theorem, d⌠⌡⎮⎮02πx()cos πxx = F (2π) - F (0) .> F(2*Pi)-F(0); − + ()cos 2 π22 π2()sin 2 π2π21π2 This looks different from the answer obtained above, but the two are actually equal> evalf(%);1.516185362> evalf(2*(2*sin(Pi^2)*cos(Pi^2)*Pi^2-1+cos(Pi^2)^2)/(Pi^2));1.516185360The apparent differance is due to round off errors. We increase the number of digits and repeat the computations:> Digits:=15; := Digits15> evalf(2*(2*sin(Pi^2)*cos(Pi^2)*Pi^2-1+cos(Pi^2)^2)/(Pi^2));1.51618535331448> evalf((cos(2*Pi^2)+2*Pi^2*sin(2*Pi^2))/(Pi^2)-1/(Pi^2));1.51618535331451> Digits:=10; := Digits 10Name: <your name goes here> Lab 8 Section: <your section goes here> > restart;Exercises 1. Let f (x) = − x35 x + 1 , -2 < x < 2. a) Partition the interval [ -2 , 2 ] into 10 equal equal parts For the ck's , use the right hand endpoints of the intervals. Plot the graph of f and the approximating rectangles. What is the value of xk - x − k 1 in this example? What is the value of c7? > > b) Partition the interval into n equal parts and again use the right hand endpoints as the ck's. Display the corresponding sum, and find its limit as → n ∞. Write an equation expressing the result as the value of an integral. > > c) Find an antiderivative of f (x), and use the Fundamental Theorem to evaluate the integral. Verify that the result is the same as in part b). > > 2. Let g (x) = x2 cos x , 0 < x < 2π. Partition [ 0 , 2π ] into n equal parts and use the left hand endpoints of the subintervals as the ck's.a) What are the values of x0 , x1, x2, and x3 ? c2? The answers will be in terms of n. > > b) Display the approximating sum corresponding to the partition, find a formula for its value, and find its limit as → n ∞. Write an equation expressing the resulting number as an integral. > > c) Find an antiderivative of g (x), and use the Fundamental Theorem to evaluate the integral. Verify that the result is the same as in part b). >
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