Efficiently Approximate ∫101F(X)ⅆx Using Left Riemann Sum With Four Subintervals

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.

To find the left Riemann sum approximation for ∫101f(x)ⅆx using the four subintervals, we need to add up the areas of four rectangles, where the height of each rectangle is the value of f at the left endpoint of the subinterval.

The left endpoints of the subintervals are 1, 2, 4, and 7. The corresponding values of f are 1.6, 2, 2.6, and 3.2, respectively.

The width of each subinterval is (7-1)/4 = 1.5.

So, the left Riemann sum approximation L is:

L = f(1)×1.5 + f(2)×1.5 + f(4)×1.5 + f(7)×1.5
= 1.6×1.5 + 2×1.5 + 2.6×1.5 + 3.2×1.5
= 16.5

Therefore, the left Riemann sum approximation for ∫101f(x)ⅆx using the four subintervals is 16.5.

Thus, the statement that is true is: The left Riemann sum approximation L for ∫101f(x)ⅆx using the four subintervals indicated by the table is 16.5.

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