Riemann Sum Right Endpoints MATH 149 Spring 2008 April 2008 Rb Suppose that J a f x dx that h b a n and that x0 a xj a jh for j 1 2 n Let the sampling points for the Riemann sum approximating the definite integral J be the right endpoints x j xj for j 1 2 n Then the Riemann sum approximation for J is simply the arithmetic mean of the right endpoints multiplied by the length of the interval a b from a to b namely b a n n n X X X RRS n f x j xj f x j h h f xj j 1 j 1 j 1 We claim that if f 0 x M for a x b then J RRS n M Z b a 2 2n xj Let Jj f x dx for j 1 2 n denote the pieces of the definite integral over each xj 1 of the sub intervals Then we may write the definite integral as a sum J J1 J2 Jn Therefore the difference between the definite integral and its approximation by a Riemann sum with sampling points being right endpoints for an equally spaced partition is J RSS n J1 hf x1 J2 hf x2 Jn hf xn n X Jj hf xj j 1 Hence J RSS n n X Jj hf xj j 1 To bound the individual terms in the sum consider the situation for a more general interval namely Z d Z d Z d Z d f x dx hf d f x dx f d dx f x f d dx d h d h d h d h By applying the Mean Value Theorem to the difference in the last integrand f x f d f 0 cx x d for some cx between x and d for each x in the closed interval d h d If we assume that the derivative of f x is bounded that is if we assume that there is a positive real number M such that for x in the interval a b the derivative satisfies f 0 x M then we get a bound on the absolute value of the integral because Z d Z d Z d f x f d dx f 0 cx x d dx f x f d dx d h d h d h Hence applying the bound on the derivative Z d Z d Z 0 f cx x d dx M x d dx d h d h d M d x dx d h because d h x d 1 Riemann Sum Right Endpoints MATH 149 Spring 2008 April 2008 But Z d d x 2 M d x dx M 2 d h Therefore Z d M d h Z d d f x f d dx d h d d 2 d d h 2 M h2 2 2 f 0 cx x d dx d h M h2 2 This result can be applied to each of the sub intervals of the partition of a b by setting d xj for the right hand endpoint and noting that then xj 1 xj h d h for the left hand endpoint Therefore Z xj M h2 Jj hf xj f x dx hf xj 2 xj 1 Adding these terms together yields the following bound on the error in approximating the definite integral with the arithmetic mean of equally spaced function values in terms of an upper bound on the derivative of the integrand J RRS n n X j 1 Jj hf xj n X M h2 2 j 1 n M h2 b a h b a 2 M M 2 2 2n since n h b a This is the result we wished to show Note 1 The preceding argument was inspired by Courant and John See Chapter 6 Numerical Methods Section 6 1 Computation of Integrals Sub section a Approximation by Rectangles pp 482 483 and 6 1 Problem 4 p 507 below Note 2 For a recent treatment of error bounds for other numerical approximations for the definite integral see the paper by Talman references Richard Courant and Fritz John Introduction to Analysis vol 1 Wiley Interscience 1965 ISBN 0 471 17860 4 Louis A Talman Simpson s rule is exact for quintics American Mathematical Monthly 113 2006 144 155 AMS LATEX 2
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