If
f(x) = 3x^2 − 2x, 0 ≤ x ≤ 3,
evaluate the Riemann sum with n = 6, taking the sample points to be right endpoints.
To evaluate the Riemann sum of a function with right endpoints, we need to divide the interval [a, b] into n equal subintervals, where a and b are the lower and upper limits of the interval respectively.
In this case, the interval is given as 0 ≤ x ≤ 3, so a = 0 and b = 3. The number of subintervals, n, is given as 6.
To calculate the width of each subinterval, Δx, we use the formula:
Δx = (b - a) / n
Substituting in the values, we have:
Δx = (3 - 0) / 6
= 3 / 6
= 0.5
Next, we need to determine the right endpoints for each subinterval. Since we have 6 subintervals, the right endpoints would be:
x1 = a + Δx
x2 = x1 + Δx
x3 = x2 + Δx
x4 = x3 + Δx
x5 = x4 + Δx
x6 = x5 + Δx
Substituting the values, we have:
x1 = 0 + 0.5
= 0.5
x2 = 0.5 + 0.5
= 1.0
x3 = 1.0 + 0.5
= 1.5
x4 = 1.5 + 0.5
= 2.0
x5 = 2.0 + 0.5
= 2.5
x6 = 2.5 + 0.5
= 3.0
Now we can evaluate the Riemann sum using the right endpoints:
Riemann Sum = f(x1)Δx + f(x2)Δx + f(x3)Δx + f(x4)Δx + f(x5)Δx + f(x6)Δx
Substituting the function f(x) = 3x^2 - 2x into the formula, we get:
Riemann Sum = [3(x1)^2 - 2(x1)]Δx + [3(x2)^2 - 2(x2)]Δx + [3(x3)^2 - 2(x3)]Δx + [3(x4)^2 - 2(x4)]Δx + [3(x5)^2 - 2(x5)]Δx + [3(x6)^2 - 2(x6)]Δx
Now, we can substitute the values of x1, x2, x3, x4, x5, x6, and Δx into the formula and calculate the Riemann sum.