Calculate Left Riemann Sum Approximation for Integral ∫101 f(x)dx using 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

To find the left Riemann sum approximation for the integral ∫101 f(x)dx using four subintervals, we need to calculate the area of the rectangles bounded by the function and the x-axis

To find the left Riemann sum approximation for the integral ∫101 f(x)dx using four subintervals, we need to calculate the area of the rectangles bounded by the function and the x-axis.

Looking at the table provided, we can see that the given values of f(x) are increasing. This means that the function is also increasing for all values of x within the interval [1, 10].

To calculate the left Riemann sum approximation, we will use the following steps:

1. Calculate the width of each subinterval:
Δx = (b – a) / n
Δx = (10 – 1) / 4
Δx = 9 / 4
Δx = 2.25

2. Calculate the left endpoint for each subinterval:
a = 1
a1 = a
a2 = a + Δx
a3 = a + 2Δx
a4 = a + 3Δx

a1 = 1
a2 = 1 + 2.25
a3 = 1 + 2(2.25)
a4 = 1 + 3(2.25)

a1 = 1
a2 = 3.25
a3 = 5.5
a4 = 7.75

3. Calculate the height (f(x)) for each left endpoint:
f(a1) = 0.4
f(a2) = 1.3
f(a3) = 2.2
f(a4) = 3.1

4. Calculate the area of each rectangle:
Rectangle 1: width * height = Δx * f(a1) = 2.25 * 0.4 = 0.9
Rectangle 2: Δx * f(a2) = 2.25 * 1.3 = 2.925
Rectangle 3: Δx * f(a3) = 2.25 * 2.2 = 4.95
Rectangle 4: Δx * f(a4) = 2.25 * 3.1 = 6.975

5. Calculate the sum of the areas of the rectangles to get the left Riemann sum approximation:
L = Rectangle 1 + Rectangle 2 + Rectangle 3 + Rectangle 4
L = 0.9 + 2.925 + 4.95 + 6.975
L = 15.75

Now, to answer the question: Which of the following statements is true?

Based on our calculations, the left Riemann sum approximation for ∫101 f(x)dx using four subintervals is L = 15.75.

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