Find the value of the right-endpoint Riemann sum in terms of n
f(x)=x^2 [0,2]
To find the value of the right-endpoint Riemann sum in terms of n for the function f(x) = x^2 over the interval [0,2], we need to follow these steps:
1. Determine the width of each rectangle. Since we are using the right-endpoint Riemann sum, the width of each rectangle is given by Δx = (b - a) / n, where n is the number of subintervals, b is the upper bound of the interval (2 in this case), and a is the lower bound of the interval (0 in this case).
Therefore, Δx = (2 - 0) / n = 2/n.
2. Find the right-endpoints of each subinterval. Starting from the lower bound (0), the right-endpoints of the subintervals are given by x_i = a + i * Δx, where i ranges from 1 to n.
In this case, the right-endpoints are as follows:
x_1 = 0 + 1 * (2/n) = 2/n
x_2 = 0 + 2 * (2/n) = 4/n
x_3 = 0 + 3 * (2/n) = 6/n
...
x_n = 0 + n * (2/n) = 2
3. Evaluate the function at each right-endpoint. In this case, we need to calculate f(x_i) = (x_i)^2 for each x_i.
f(x_1) = (2/n)^2 = (4/n^2)
f(x_2) = (4/n)^2 = (16/n^2)
f(x_3) = (6/n)^2 = (36/n^2)
...
f(x_n) = (2)^2 = 4
4. Calculate the sum of the function values times the width of each rectangle. This is the right-endpoint Riemann sum.
R_n = (4/n^2) * (2/n) + (16/n^2) * (2/n) + (36/n^2) * (2/n) + ... + 4 * (2/n)
= (4 * 2 / n^3) * (1 + 4 + 9 + ... + (n-1)^2 + n)
= (8 / n^3) * (1^2 + 2^2 + 3^2 + ... + (n-1)^2 + n^2)
So, the value of the right-endpoint Riemann sum in terms of n for the function f(x) = x^2 over the interval [0,2] is (8 / n^3) * (1^2 + 2^2 + 3^2 + ... + (n-1)^2 + n^2).
To find the value of the right-endpoint Riemann sum in terms of n for the function f(x) = x^2 on the interval [0,2], we need to divide the interval into n subintervals of equal width and evaluate the function at the right endpoint of each subinterval.
The width of each subinterval Δx is equal to (b - a) / n, where a and b are the endpoints of the interval. In this case, a = 0 and b = 2, so Δx = (2 - 0) / n = 2/n.
The right endpoints of the subintervals are given by the values x_i = a + i * Δx, where i ranges from 1 to n. In this case, a = 0, so x_i = i * Δx = i * (2/n).
To evaluate the function at the right endpoints, substitute x = i * (2/n) into the function f(x) = x^2. Therefore, the value of the right-endpoint Riemann sum in terms of n is:
R_n = Σ f(i * (2/n)) * Δx
where Σ represents the summation from i = 1 to n.
In this case, the function is f(x) = x^2 and the width of each subinterval is Δx = 2/n.
So, the right-endpoint Riemann sum can be calculated as:
R_n = Σ (i * (2/n))^2 * (2/n)
Hope this helps!