Express the following integral as a limit of Riemann sums:

Question

BlackTom AI · Accepted Answer

The given problem statement is missing the actual answer choices, so I cannot analyze options as requested. Instead, here is a thorough, stand-alone guide to expressing a definite integral as a limit of Riemann sums, plus a concrete template you can adapt once you have the specific f, a, b, and any sample points.

General idea
- For a definite integral ∫_a^b f(x) dx, partition the interval [a, b] into n subintervals of equal width: Δx = (b − a)/n.
- Choose a sample point x_i* in each subinterval [x_{i−1}, x_i], where x_i = a + iΔx and x_0 = a. Common choices are left endpoints x_i* = x_{i−1}, right endpoints x_i* = x_i, or midpoints x_i* = (x_{i−1} + x_i)/2.
- Form the Riemann sum S_n = ∑_{i=1}^n f(x_i*) Δx.
- The integral is the limit of these sums as n → ∞, provided f is integrable on [a, b]. That is, ∫_a^b f(x) dx = lim_{n→∞} ∑_{i=1}^n f(x_i*) Δx.

Step-by-step template you can apply immediately
1) Identify the components: a, b, and the function f(x) given in the problem (even if not stated explicitly here). 
2) Decide on a partition: Δx = (b − a)/n. Compute the endpoints x_0 = a, x_1 = a + Δx, ..., x_n = b. 
3) Pick a sample point x_i* in each subinterval. If you aren’t told which to use, you can show the general form with a placeholder x_i*, or present the three common choices:
   - Left endpoint: x_i* = x_{i−1}
   - Right endpoint: x_i* = x_i
   - Midpoint: x_i* = (x_{i−1} + x_i)/2
4) Write the Riemann sum: S_n = ∑_{i=1}^n f(x_i*) Δx, with Δx as above and x_i* chosen per step 3.
5) State the limit: ∫_a^b f(x) dx = lim_{n→∞} S_n. Briefly justify why this holds (e.g., f is continuous on [a,b], or f is Riemann-integrable on [a,b]).

Worked generic example (structure you can fill in with your specific function)
- Suppose your integral is ∫_a^b f(x) dx with a given f. Then:
  Δx = (b − a)/n, x_i = a + iΔx.
  A typical Riemann sum is S_n = ∑_{i=1}^n f(x_i*) Δx, where x_i* ∈ [x_{i−1}, x_i].
  The limit expression is ∫_a^b f(x) dx = lim_{n→∞} ∑_{i=1}^n f(x_i*) Δx.
- If you use left endpoints: S_n,left = ∑_{i=1}^n f(x_{i−1}) Δx. The limit as n→∞ of S_n,left, when f is continuous, equals ∫_a^b f(x) dx.
- If you use right endpoints: S_n,right = ∑_{i=1}^n f(x_i) Δx. This also converges to the integral under the same conditions.
- If you use midpoints: S_n,mid = ∑_{i=1}^n f((x_{i−1} + x_i)/2) Δx. Midpoint rule often has better convergence properties for smooth f.

Notes and tips
- If the problem specifies a particular form for f or a and b, plug those in and you can write the explicit S_n and then discuss the limit. If the problem asks for a particular limit form, include the exact S_n expression corresponding to the chosen sample points.
- If f is not continuous everywhere, you still can have convergence for integrable f, but you may need a more careful justification (e.g., Darboux or Riemann integrability conditions).
- When answering with multiple-choice options later, be explicit about which choice matches the S_n expression you derived, and explain why other options (e.g., using an incorrect Δx, or wrong x_i*) fail to produce the limit of the Riemann sums for ∫_a^b f(x) dx.

Express the following integral as a limit of Riemann sums: 单项选择题

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