5. Given f [a, b] R and g [a,b] - R are both bounded...

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5. Given f [a, b] → R and g [a,b] - → R are both bounded with g(x) ≤ f(x) for all xe [a, b]. Let P be any partition of [a, b]. Prove that U(g, P) ≤U(f, P). 6. Let f [1,5] → R with f(x) = = 2x for 1 ≤ x < 3, f(3) = 8 and f(x) = -3 for 3 < x < 5. a. Use the A-R theorem to prove directly that f is integrable. b. Use the A-R theorem to calculate f 5. Given f [a, b] → R and g [a,b] - → R are both bounded with g(x) ≤ f(x) for all xe [a, b]. Let P be any partition of [a, b]. Prove that U(g, P) ≤U(f, P). 6. Let f [1,5] → R with f(x) = = 2x for 1 ≤ x < 3, f(3) = 8 and f(x) = -3 for 3 < x < 5. a. Use the A-R theorem to prove directly that f is integrable. b. Use the A-R theorem to calculate f

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b To calculate the definite integral of fx over 1 5 using the AR theorem we need to evaluate the upp... View the full answer