If

f(x) = 3x2 − 2x, 0 ≤ x ≤ 3,
evaluate the Riemann sum with n = 6, taking the sample points to be right endpoints.
R6 =

To evaluate the Riemann sum with n = 6 and right endpoints, we need to divide the interval [0, 3] into 6 subintervals of equal width.

First, let's find the width of each subinterval:
Width of each subinterval = (b - a) / n
= (3 - 0) / 6
= 3/6
= 1/2

Next, we will determine the right endpoint of each subinterval:
Right endpoint = a + (width * i), where i is the index of the subinterval starting from 1.

Using these formulas, we can calculate the right endpoints for the 6 subintervals:
Right endpoints = {0 + (1/2 * 1), 0 + (1/2 * 2), 0 + (1/2 * 3), 0 + (1/2 * 4), 0 + (1/2 * 5), 0 + (1/2 * 6)}
= {1/2, 1, 3/2, 2, 5/2, 3}

Now, we need to evaluate the function f(x) = 3x^2 - 2x at each of these right endpoints:
f(1/2) = 3(1/2)^2 - 2(1/2)
= 3/4 - 1
= -1/4

f(1) = 3(1^2) - 2(1)
= 3 - 2
= 1

f(3/2) = 3(3/2)^2 - 2(3/2)
= 27/4 - 3
= 3/4

f(2) = 3(2^2) - 2(2)
= 12 - 4
= 8

f(5/2) = 3(5/2)^2 - 2(5/2)
= 75/4 - 5
= 55/4

f(3) = 3(3^2) - 2(3)
= 27 - 6
= 21

Finally, we can calculate the Riemann sum by summing up the areas of the rectangles:
R6 = (width * (f(1/2) + f(1) + f(3/2) + f(2) + f(5/2) + f(3)))
= (1/2 * (-1/4 + 1 + 3/4 + 8 + 55/4 + 21))
= (1/2 * (59/4 + 8 + 3))
= (1/2 * (59/4 + 35/4))
= (1/2 * (94/4))
= (1/2 * 47/2)
= (47/4)
= 11.75

Therefore, the Riemann sum with n = 6 and right endpoints is 11.75.

To evaluate the Riemann sum with n = 6, taking the sample points to be right endpoints, we need to divide the interval [0, 3] into 6 equal subintervals. Then, we evaluate the function at the right endpoint of each subinterval, multiply the function value by the width of the subinterval, and sum up these products to get the Riemann sum.

The width of each subinterval can be calculated by dividing the length of the interval by the number of subintervals.

Length of interval = (upper limit - lower limit) = (3 - 0) = 3
Number of subintervals = n = 6

Width of each subinterval = (Length of interval) / (Number of subintervals) = 3 / 6 = 0.5

Now, we need to identify the right endpoints of each subinterval. Since we are using right endpoints as sample points, the right endpoint of each subinterval will be the endpoint of the subinterval or the starting point of the next subinterval.

Subinterval 1: [0, 0.5] - Right endpoint: 0.5
Subinterval 2: [0.5, 1] - Right endpoint: 1
Subinterval 3: [1, 1.5] - Right endpoint: 1.5
Subinterval 4: [1.5, 2] - Right endpoint: 2
Subinterval 5: [2, 2.5] - Right endpoint: 2.5
Subinterval 6: [2.5, 3] - Right endpoint: 3

Now we can evaluate the function at each of these right endpoints and calculate the Riemann sum by multiplying the function value by the width of each subinterval and summing up the products.

R6 = f(0.5)*0.5 + f(1)*0.5 + f(1.5)*0.5 + f(2)*0.5 + f(2.5)*0.5 + f(3)*0.5

To evaluate the function f(x) = 3x^2 - 2x at each of these right endpoints, we substitute the values of x into the function.

R6 = (3(0.5)^2 - 2(0.5))*0.5 + (3(1)^2 - 2(1))*0.5 + (3(1.5)^2 - 2(1.5))*0.5 + (3(2)^2 - 2(2))*0.5 + (3(2.5)^2 - 2(2.5))*0.5 + (3(3)^2 - 2(3))*0.5

Simplifying the expression:

R6 = (0.75 - 1)*0.5 + (3 - 2)*0.5 + (6.75 - 4.5)*0.5 + (12 - 4)*0.5 + (18.75 - 5.25)*0.5 + (27 - 6)*0.5

R6 = (0.75 - 1)*0.5 + (3 - 2)*0.5 + (6.75 - 4.5)*0.5 + (12 - 4)*0.5 + (18.75 - 5.25)*0.5 + (27 - 6)*0.5
= -0.25*0.5 + 1*0.5 + 2.25*0.5 + 8*0.5 + 13.5*0.5 + 21*0.5
= -0.125 + 0.5 + 1.125 + 4 + 6.75 + 10.5
= 22.75

Therefore, the Riemann sum R6 with the sample points as right endpoints is equal to 22.75.

Please follow directions.

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There are lots of good Riemann Sum online calculators. This one is easy, though.

The interval widths are 0.5, so all you have to do is evaluate

0.5(f(.5)+f(1.0)+...+f(3.0))