MATH 149 LABORATORY ASSIGNMENT 8 THE FUNDAMENTAL THEOREM OF CALCULUS In class we defined the definite integral of the function f x on the interval a b as b f x dx Lim a n f c x k k xk 1 k 1 where a x0 x1 x2 xn 1 xn b is a partition of a b xk 1 ck xk for k 1 n and the limit is taken as n and the lengths of the subintervals xk 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 that b f x dx F b F a 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 x 0 x 2 We will use partitions in which the xk s are equally spaced and each ck is the midpoint of xk 1 xk First we illustrate the approximating rectangles using a partition with 20 points with student f x x cos Pi x f x x cos x middlebox f x x 0 2 Pi 20 The corresponding approximating sum is middlesum f x x 0 2 Pi 20 2 1 i 19 1 1 1 i cos 2 10 10 i 0 10 2 or value 2 3 3 2 1 2 7 7 2 1 cos cos cos cos 10 20 20 20 20 4 4 20 20 1 9 2 11 11 2 13 13 2 3 3 2 cos cos cos cos 20 20 20 20 20 20 4 4 9 17 2 19 19 2 21 21 2 23 23 2 cos cos cos cos 20 20 20 20 20 20 20 20 17 5 2 27 27 2 29 29 2 31 31 2 cos cos cos cos 4 4 20 20 20 20 20 20 5 33 2 7 7 2 37 37 2 39 39 2 cos cos cos cos 20 20 4 4 20 20 20 20 evalf 33 1 582743036 If we use a partition with n points the approximating sum is middlesum f x x 0 2 Pi n 1 2 2 i 1 2 2 i cos n 1 n 2 2 n i 0 n MAPLE knows a formula for this sum value 2 2 2 2 2 2 cos sin 1 cos 2 sin sin 2 cos 2 n n n n 2 2 2 2 2 cos 1 cos 1 n n 2 2 2 cos cos n 2 2 n cos 1 n 2 2 2 2 2 cos sin 1 cos n n n n 2 2 cos 1 n n 2 2 n cos 1 n 2 cos n 2 We now take the limit as n this will be the exact value of x cos x dx 0 limit n infinity 2 2 cos 2 4 sin cos 2 2 2 2 2 evalf 1 516185360 Next we use MAPLE to find an antiderivative F x for f x The command is int f x x cos x x sin x NOTE THAT MAPLE DOES NOT INCLUDE THE C F x cos Pi x Pi x sin Pi x Pi 2 2 F x cos x x sin x You 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 x will be developed in Math 152 2 2 By the Fundamental Theorem x cos x dx F 2 F 0 0 F 2 Pi F 0 cos 2 2 sin 2 2 2 2 1 This looks different from the answer obtained above but the two are actually equal evalf 2 2 1 516185362 evalf 2 2 sin Pi 2 cos Pi 2 Pi 2 1 cos Pi 2 2 Pi 2 1 516185360 The apparent differance is due to round off errors We increase the number of digits and repeat the computations Digits 15 Digits 15 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 10 Name your name goes here Lab 8 Section your section goes here restart Exercises 1 Let f x x3 5 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 xk 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 2 Partition 0 2 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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