Find a Riemann Sum formula for evaluation points
that are one-third of the way from the left endpoint to the right endpoint.
so divide the limits by n evenly spaced increments, then each f(x) will be evaluated at each 1/3*increment + xleft.
I am not certain what I can do for you more than this.
So, with limits a and b, that will be
n-1
∑ f(a+(i + 1/3)*(b-a)/n)*(b-a)/n
i=0
To find a Riemann Sum formula for evaluation points that are one-third of the way from the left endpoint to the right endpoint, we need to consider the partition of the interval [a, b] into n sub-intervals.
Let's denote the length of each sub-interval as Δx, which is given by Δx = (b - a) / n. Now, we want the evaluation points to be one-third of the way from the left endpoint to the right endpoint within each sub-interval.
The left endpoint of each sub-interval can be found by using the formula a + iΔx, where i represents the index of the sub-interval starting from 0. To find the evaluation points one-third of the way from the left endpoint to the right endpoint, we can use the formula:
x_i = a + iΔx + (1/3)Δx,
where x_i represents the ith evaluation point.
Now, using these evaluation points, the Riemann Sum formula can be given as:
R_n = Σ[f(x_i)Δx],
where Σ represents the summation from i=0 to n-1, and f(x_i) represents the function evaluated at x_i.
In summary, to find a Riemann Sum formula for evaluation points that are one-third of the way from the left endpoint to the right endpoint, we use the formula x_i = a + iΔx + (1/3)Δx to determine the evaluation points and then use the Riemann Sum formula R_n = Σ[f(x_i)Δx] to evaluate the approximation.