Evaluate the Riemann sum for (x) = x3 − 6x, for 0 ≤ x ≤ 3 with six subintervals, taking the sample points, xi, to be the right endpoint of each interval and please explain, using a graph of f(x), what the Riemann sum represents.
My cousin needs help with her homework but I have no idea what this stuff is about. If someone could please help us and explain the work so that I can maybe try and remember how to help her, that would be nice :)
To evaluate the Riemann sum for the function f(x) = x^3 - 6x over the interval [0, 3] with six subintervals, we need to follow these steps:
Step 1: Determine the width of each subinterval.
Since we have six subintervals over the interval [0, 3], the width of each subinterval is given by:
Δx = (b - a) / n
Δx = (3 - 0) / 6
Δx = 1/2
Step 2: Determine the right endpoints of each subinterval.
The right endpoints of the subintervals can be found by adding the width of each subinterval to the left endpoint (0 in this case). Using the previously calculated width, we can determine the right endpoints as:
x1 = 0 + Δx = 0 + 1/2 = 1/2
x2 = x1 + Δx = 1/2 + 1/2 = 1
x3 = x2 + Δx = 1 + 1/2 = 3/2
x4 = x3 + Δx = 3/2 + 1/2 = 2
x5 = x4 + Δx = 2 + 1/2 = 5/2
x6 = x5 + Δx = 5/2 + 1/2 = 3
Step 3: Compute the value of the function at each right endpoint.
Evaluating the function f(x) = x^3 - 6x at each of these right endpoints, we get:
f(x1) = f(1/2) = (1/2)^3 - 6(1/2) = -23/8
f(x2) = f(1) = (1)^3 - 6(1) = -5
f(x3) = f(3/2) = (3/2)^3 - 6(3/2) = -27/8
f(x4) = f(2) = (2)^3 - 6(2) = -8
f(x5) = f(5/2) = (5/2)^3 - 6(5/2) = -45/8
f(x6) = f(3) = (3)^3 - 6(3) = -9
Step 4: Compute the Riemann sum.
Finally, the Riemann sum is given by summing the areas of rectangles with width Δx and height f(xi) for each right endpoint xi:
Riemann sum = Δx * (f(x1) + f(x2) + f(x3) + f(x4) + f(x5) + f(x6))
= (1/2) * (-23/8 - 5 - 27/8 - 8 - 45/8 - 9)
= (1/2) * (-23/8 - 40/8 - 27/8 - 72/8 - 45/8 - 72/8)
= (1/2) * (-279/8)
= -279/16
To help explain what the Riemann sum represents, let's graph the function f(x) = x^3 - 6x over the interval [0, 3].
The Riemann sum represents the approximation of the area underneath the curve of the function f(x) over the given interval using a series of rectangles. The width of each rectangle is determined by dividing the interval into equal subintervals, and the height of each rectangle is determined by evaluating the function at the right endpoint of each subinterval.
By summing the areas of these rectangles, we can approximate the total area underneath the curve. In this case, the Riemann sum evaluates the sum of these rectangle areas for the function f(x) = x^3 - 6x, over the interval [0, 3], with six subintervals where the right endpoint of each subinterval is used as the sample point. The resulting Riemann sum value, -279/16, represents an approximation of the total area underneath the curve of the function.
break the interval into six parts,
zero to .5, .5 to 1, etc.
then calculate the function in those intervals, you can choose the left number, the right number, or the center number. Lets do it on the RIGHT endpoint
first calculate f(.5), then f(1), and so on.
Each of those values are multiplied by the width of the interval, in our case .5.
Now take those six areas (width x height), add them.