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Let ln denote the left-endpoint sum using n subintervals. Compute the indicated left sum for the given function on the indicated interval. (Round your answer to four decimal places.) l6 for f(x) = 1 x(x - 1) on [3, 6].

User Emira
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Answer:

The left sum l_6 for the given function on the interval [3, 6] with 6 subintervals is 0.2591.

Explanation:

Computing the Left Sum l_6

To compute the left sum l_6 for the function f(x) = 1 / (x(x - 1)) on the interval [3, 6] with 6 subintervals, we follow these steps:

1. Divide the interval into subintervals:

The interval [3, 6] is divided into 6 subintervals of equal width. The width of each subinterval is:

Δx = (b - a) / n = (6 - 3) / 6 = 0.5

Therefore, the subintervals are:

[3, 3.5], [3.5, 4], [4, 4.5], [4.5, 5], [5, 5.5], [5.5, 6]

2. Evaluate the function at the left endpoint of each subinterval:

We evaluate the function f(x) at the left endpoint (x_i) of each subinterval:

f(3) = 1 / (3 * 2) = 1 / 6

f(3.5) = 1 / (3.5 * 2.5) = 2 / 8.75

f(4) = 1 / (4 * 3) = 1 / 12

f(4.5) = 1 / (4.5 * 3.5) = 2 / 15.75

f(5) = 1 / (5 * 4) = 1 / 20

f(5.5) = 1 / (5.5 * 4.5) = 2 / 24.75

3. Multiply the function values by the subinterval width:

We multiply each function value by the subinterval width Δx = 0.5:

Δx * f(3) = 0.5 * 1 / 6 = 0.0833

Δx * f(3.5) = 0.5 * 2 / 8.75 = 0.1143

Δx * f(4) = 0.5 * 1 / 12 = 0.0417

Δx * f(4.5) = 0.5 * 2 / 15.75 = 0.0635

Δx * f(5) = 0.5 * 1 / 20 = 0.0250

Δx * f(5.5) = 0.5 * 2 / 24.75 = 0.0404

4. Add the products to find the left sum:

Finally, we add the products of each function value and subinterval width to get the left sum l_6

l_6 = 0.0833 + 0.1143 + 0.0417 + 0.0635 + 0.0250 + 0.0404 ≈ 0.2591

Therefore, the left sum l_6 for the given function on the interval [3, 6] with 6 subintervals is approximately 0.2591.

User TomDane
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