Analyzing Riemann Sums for Integral Calculation
The question asks which of the given limits of a Riemann sum represents the value of the integral ∫ from -2 to 2 of (x^3 - 4x) dx.
For a Riemann sum to represent the integral of a function, it must be in the form of the limit as n approaches infinity of the sum from k=1 to n of f(x_k)Δx, where Δx is the width of each sub-interval, and x_k is a sample point in the k-th sub-interval.
Now, because the interval of integration is from -2 to 2, the total width of the interval is 4. Therefore, Δx = (b-a)/n = (2 - (-2))/n = 4/n.
Next, we need to pick the sample points. For problems like this, we often use either left endpoints, right endpoints, or midpoints. Since the problem does not specify, we'll examine each of the options provided to see which one matches up.
Looking at the options, the sample points x_k should be something like -2 + k(Δx) or -2 + (k - 1/2)(Δx) or -2 + (k-1)(Δx) for the left, midpoint, and right Riemann sums respectively.
Let's examine the structure of the Riemann sums in the options:
(A) x_k appears to be -2 + (3/2)(k/n) which is not of the correct form for any of the typical Riemann sum sample points.
(B) x_k appears to be -2 + (k/n) which is not correct because it lacks the width of the sub-interval, Δx, in the multiplication with k.
(C) x_k appears to be -2 + (5/2)(k/n) which is again not of the correct form for the Riemann sum sample points.
(D) The sum in this option has not boxed or circled any term within it to identify x_k.
Since options A, B, and C have clear structural issues with their x_k values, none of these correspond to the correct form for a Riemann sum for the integral given.
Now, let's re-evaluate option D. The width of each sub-interval is (4/n) and when we plug in k as our index we should get:
x_k = -2 + (k-1)(4/n) for the left Riemann sum
or
x_k = -2 + k(4/n) for the right Riemann sum
or
x_k = -2 + (4/n)(k - 1/2) for the midpoint Riemann sum.
The expression in option D suggests that it's probably intending the midpoint Riemann sum, because it has an additional 1/2 term in the representation of x_k (since it doesn't fit the left or right endpoint patterns).
Therefore, let's confirm the structure of x_k for the midpoint Riemann sum by expanding the expression used for x_k in option D:
x_k = -2 + (2 + 5/2k/n) = -2 + (2 + (5k)/(2n))
This is not immediately recognizable as a midpoint sum expression. To match the desired structure, we should see:
x_k = -2 + (4/n)(k - 1/2) = -2 + (4k/n) - (2/n)
However, it's evident that the expressions in option D do not match the form of the midpoint Riemann sum after manipulating the terms.
Therefore, we need to correct the expression provided:
The midpoint for each sub-interval would be at x_k = -2 + (4/n)(k - 1/2) which simplifies to
x_k = -2 + (4k/n) - (2/n)
And we would then construct the sum as:
Σ_{k=1}^n f(x_k)Δx
It seems that none of the options A-D perfectly match the expected structure for a Riemann sum that would calculate the integral ∫ from -2 to 2 of (x^3 - 4x) dx. Option D is the closest, but the form of x_k provided is not correct for a typical midpoint Riemann sum, assuming that was the intention. If none of the options matches, an error may be present in the formulation of the answers, or additional context may be required to determine the correct response.
