Basic Properties of the Integral Notes

Note that b is now the lower limit on the integral

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Unformatted text preview: 1 5 3 3 5 1 7 b−a 7 = f a+ b +f a+ b +f a+ b +f a+ b 8 8 8 8 8 8 8 8 4 b−a 4 f a+ (2) a We’re now going to write out the midpoint Riemann sum approximation to b f (x) dx with 4 subintervals. Note that b is now the lower limit on the integral and a is now the upper limit on the integral. This is likely to cause confusion when we write out the Riemann sum, 3 January 13, 2014 so we’ll temporarily rename b to A and a to B . The midpoint Riemann sum approximation B to A f (x) dx with 4 subintervals is 3B −A 5B −A 1B −A +f A+ +f A+ +f 24 24 24 1 3 5 5 3 1 7 = f A+ B +f A+ B +f A+ B +f A+ 8 8 8 8 8 8 8 f A+ A+ 7 B 8 7B −A 24 B−A 4 B−A 4 Now setting A = b and B = a, we have that the midpoint Riemann sum approximation to a f (x) dx with 4 subintervals is b f 7 1 5 3 3 5 1 7 b+ a +f b+ a +f b+ a +f b+ a 8 8 8 8 8 8 8 8 a−b 4 (3) The curly brackets in (2) and (3) are equal to each other — the terms are just in the reverse order. So (3)= −(2). The same computation with n subintervals shows that the midpoint a b Riemann sum approximations to b f (x) dx and a f (x) dx with n subintervals are negatives a b of each other. Taking the limit n → ∞ gives b f (x) dx = − a f (x) dx. Example 4 Recall that if x ≥ 0 x |x| = −x if x ≤ 0 So 1 −1 0 f |x| dx = −1 0 = −1 1 f |x| dx + f |x| dx 0 1 f (−x) dx + f (x) dx 0 Example 4 4 January 13, 2014 Theorem 5 (Inequalities for Integrals). Let a ≤ b be real numbers and let the functions f (x) and g (x) be integrable on the interval a ≤ x ≤ b. (a) If f (x) ≥ 0 for all a ≤ x ≤ b, then a b f (x) dx ≥ 0 (b) If there are constants m and M such that m ≤ f (x) ≤ M for all a ≤ x ≤ b, then a m(b − a) ≤ f (x) dx ≤ M (b − a) b (c) If f (x) ≤ g (x) for all a ≤ x ≤ b, then a f (x) dx ≤ b (d) We have b g (x) dx a a b a f (x) dx ≤ |f (x)| dx b Proof. (a) Just says that if the curve y = f (x) lies above...
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This note was uploaded on 02/12/2014 for the course MATH 101 taught by Professor Broughton during the Spring '08 term at The University of British Columbia.

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