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.

MATH_1225_64262_202506 5.2 The Definite Integral

Express the following integral as a limit of Riemann sums:

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Match the type of estimation technique shown in the pictures with the correct label. 1: 2: 3: 4:

Find the difference between the upper and lower estimates of the distance traveled at velocity 25−t2 on the interval 1≤t≤4 for 1000 subdivisions.

Suppose we want to find the area under the curve (pictured below) on the interval using a Riemann sum with rectangles. Which of the following provides the most accurate approximation of the area under the curve on the given interval?

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You should see 2 equations in this question. Does the 2nd equation render correctly in your browser? It should look the same as the image in this box: ∫ 0 1 𝑥 2 𝑑 𝑥 = lim 1 𝑛 ( 1 2 𝑛 2 + 2 2 𝑛 2 + . . . + ( 𝑛 2 𝑛 2 ) ) = lim ∑ 𝑛 𝑚 𝑥 × 𝑛 2 2 3

You should see 2 equations in this question. Does the 2nd equation render correctly in your browser? It should look the same as the image in this box: ∫ 1 0 x2dx=lim 1 n ( 12 n2 + 22 n2 +...+( n2 n2 ))=lim∑ n m x × n2 23

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MATH_1225_64262_202506 5.2 The Definite Integral

Express the following integral as a limit of Riemann sums:

查看解析

登录即可查看完整答案

类似问题

Match the type of estimation technique shown in the pictures with the correct label. 1: ____ 2: ____ 3: ____ 4: ____

Find the difference between the upper and lower estimates of the distance traveled at velocity 25−t2 on the interval 1≤t≤4 for 1000 subdivisions.

Suppose we want to find the area under the curve (pictured below) on the interval using a Riemann sum with rectangles. Which of the following provides the most accurate approximation of the area under the curve on the given interval?

Suppose we want to find the area under the curve (pictured below) on the interval using a Riemann sum with rectangles. Which of the following provides the most accurate approximation of the area under the curve on the given interval?

Suppose represents the velocity of an object on an interval . Using the Riemann sum approximation, , to approximate the area under the curve estimates which of the following?

You should see 2 equations in this question. Does the 2nd equation render correctly in your browser? It should look the same as the image in this box: ∫ 0 1 𝑥 2 𝑑 𝑥 = lim 1 𝑛 ( 1 2 𝑛 2 + 2 2 𝑛 2 + . . . + ( 𝑛 2 𝑛 2 ) ) = lim ∑ 𝑛 𝑚 𝑥 × 𝑛 2 2 3

You should see 2 equations in this question. Does the 2nd equation render correctly in your browser? It should look the same as the image in this box: ∫ 1 0 x2dx=lim 1 n ( 12 n2 + 22 n2 +...+( n2 n2 ))=lim∑ n m x × n2 23

更多留学生实用工具

希望你的学习变得更简单

Match the type of estimation technique shown in the pictures with the correct label. 1: 2: 3: 4: