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.