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Question

x | 2 | 4 | 6 | 7 | 11

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f(x) | 0.7 | 1.1 | 1.9 | 2.5 | 5.3

The continuous function f is known to be increasing for all x. Selected values of f are given in the table above. Let L be the left Riemann sum approximation for (top value=11)∫(bottom value=2) f(x) dx using the four subintervals indicated by the table. Which of the following statements is true?

a) L=13.95 and is an underestimate for 11∫2f(x)dx.

b) L=13.95 and is an overestimate for 11∫2f(x)dx.

c) L=15.5 and is an underestimate for 11∫2f(x)dx.

d) L=15.5 and is an overestimate for 11∫2f(x)dx.

Answer 1

L = f(2)(4-2) + f(4)(6-4) + f(6)(7-6) + f(7)(11-7) = 0.7(2) + 1.1(2) + 1.9(1) + 2.5(4) = 15.5

Since f(x) is increasing on the entire interval, L is an underestimate of the actual value of the integral.