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 ∫101f(x)ⅆx using the four subintervals indicated by the table. Which of the following statements is true
L=2.8 and is an underestimate for ∫101f(x)ⅆx∫110f(x)ⅆx.
First, we have to calculate the width of subintervals:
b-a / n = (120-101)/4 = 4.75
Then we can calculate the left Riemann sum:
L = f(a)*Δx + f(a+Δx)*Δx + f(a+2Δx)*Δx + f(a+3Δx)*Δx
L = f(101)*4.75 + f(105.75)*4.75 + f(110.5)*4.75 + f(115.25)*4.75
L ≈ 1263.875
Therefore, the approximation of the integral using the left Riemann sum with four subintervals is approximately 1263.875.
Since the function f is known to be increasing for all x, its left Riemann sum would be an underestimate of the actual integral. Thus, the true value of the integral would be greater than the approximation L. Therefore, the statement (D) L ≤ ∫101f(x)ⅆx is false.
The correct answer is (B) L < ∫101f(x)ⅆx.
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