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
To calculate the left Riemann sum approximation for ∫101f(x)ⅆx using the given table, we need to consider the four subintervals indicated
To calculate the left Riemann sum approximation for ∫101f(x)ⅆx using the given table, we need to consider the four subintervals indicated.
From the table, we have the following values:
x = 1, f(x) = 2
x = 3, f(x) = 4
x = 6, f(x) = 6
x = 10, f(x) = 10
The left Riemann sum approximation can be calculated using the formula:
L = ∑[f(x)*Δx]
Where L represents the left Riemann sum, f(x) represents the function values, and Δx represents the width of each subinterval.
First, we need to calculate the width of each subinterval:
Δx = (b – a) / n
Where b is the upper limit of integration (10 in this case), a is the lower limit (1 in this case), and n is the number of subintervals (4 in this case).
Δx = (10 – 1) / 4 = 2.25
Now, we can calculate the left Riemann sum approximation:
L = f(1) * Δx + f(3) * Δx + f(6) * Δx + f(10) * Δx
Substituting the given values:
L = 2 * 2.25 + 4 * 2.25 + 6 * 2.25 + 10 * 2.25
L = 4.5 + 9 + 13.5 + 22.5
L = 49.5
Therefore, the left Riemann sum approximation for ∫101f(x)ⅆx using the four subintervals indicated by the table is 49.5.
Based on this calculation, the correct statement on which of the following is true is that the left Riemann sum approximation for ∫101f(x)ⅆx using the four subintervals indicated is 49.5.
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