Evaluate integrate (x ^ 3 - 6x) dx from 0 to 3 taking the sample points to be right endpoints.

To evaluate the integral of the function f(x) = x^3 - 6x from 0 to 3 using right endpoints, we first need to understand that this is a definite integral problem and we are being asked to use the Riemann sum approximation method.

Step 1: Define the function The function given is f(x) = x^3 - 6x.

Step 2: Define the interval The interval given is [0, 3].

Step 3: Define the number of subintervals In this case, we are not given a specific number of subintervals, but we are told to use right endpoints. This means that we will use the value of the function at the right endpoint of each subinterval to determine the height of the rectangle in the Riemann sum.

Step 4: Calculate the width of each subinterval The width of each subinterval (Δx) is given by (b - a)/n, where a and b are the limits of integration and n is the number of subintervals. In this case, Δx = (3 - 0)/3 = 1.

Step 5: Calculate the Riemann sum The Riemann sum is given by the sum of the areas of the rectangles, which is the sum of the function values at the right endpoints of the subintervals times the width of the subintervals. In this case, the Riemann sum is:

f(1)Δx + f(2)Δx + f(3)Δx = [(1^3 - 61) + (2^3 - 62) + (3^3 - 63)] * Δx = [-5 + -4 + 9] * 1 = 0

So, the approximate value of the integral of f(x) from 0 to 3 using right endpoints is 0.

Please note that this is an approximation. The exact value of the integral can be found using the Fundamental Theorem of Calculus, which would give a different result.