Final answer:
The left-endpoint (Ln) and corresponding right-endpoint (Rn) sums are methods of approximating the definite integral of a function over an interval. To calculate Ln and Rn for the function f(x) = 1/x(x-1) on the interval [4,7] using 6 subintervals, we divide the interval into subintervals and sum the left and right endpoints of each subinterval multiplied by the width. We then evaluate the function at each point to get the values of Ln and Rn.
Step-by-step explanation:
The left-endpoint sum, denoted as Ln, is a method used to approximate the definite integral of a function over a specific interval by partitioning the interval into smaller subintervals and summing the left endpoints of each subinterval multiplied by the width of the subinterval. The corresponding left-endpoint sum, denoted as Rn, follows the same method, but uses right endpoints instead of left endpoints.
For the function f(x) = 1/x(x-1) on the interval [4,7], we need to calculate the left-endpoint sum L6 and the corresponding right-endpoint sum R6 using 6 subintervals.
To calculate L6, we need to find the width of each subinterval, which is (b-a)/n = (7-4)/6 = 3/6 = 1/2. We then calculate the left-endpoint of each subinterval by plugging in the x-coordinate of the left endpoint into the function. The left endpoints are 4, 4.5, 5, 5.5, 6, and 6.5. We evaluate the function at each of these points and sum up their values multiplied by the width. That is:
L6 = 1/2 [f(4) + f(4.5) + f(5) + f(5.5) + f(6) + f(6.5)]
Similarly, to calculate R6, we find the right endpoints of each subinterval by plugging in the x-coordinate of the right endpoint into the function. The right endpoints are 4.5, 5, 5.5, 6, 6.5, and 7. We evaluate the function at each of these points and sum up their values multiplied by the width. That is:
R6 = 1/2 [f(4.5) + f(5) + f(5.5) + f(6) + f(6.5) + f(7)]