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

Question

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

BlackTom AI · Accepted Answer

The Riemann-sum representation of the integral is the key concept: a definite integral over [0,1] equals the limit of Riemann sums. For f(x)=x^2 a standard right-endpoint Riemann sum with n subintervals is Δx = 1/n and x_k = k/n, so the sum is (1/n) Σ_{k=1}^n (k/n)^2. This is exactly the second equation in the image.

Step-by-step reasoning and calculations:
1. Write the Riemann sum form shown: (1/n) Σ_{k=1}^n (k/n)^2 = (1/n^3) Σ_{k=1}^n k^2.
2. Use the formula for the sum of squares: Σ_{k=1}^n k^2 = n(n+1)(2n+1)/6.
3. Substitute: (1/n^3) × n(n+1)(2n+1)/6 = (n(n+1)(2n+1)) / (6 n^3).
4. Simplify: = ((n+1)(2n+1)) / (6 n^2).
5. Take the limit as n → ∞: leading terms give (2n^2)/(6 n^2) = 2/6 = 1/3.

Confirm by direct integration: ∫_0^1 x^2 dx = [x^3/3]_0^1 = 1/3, matching the limit of the Riemann sums above. Thus the displayed second equation correctly presents the Riemann-sum expansion whose limit equals the integral.

Why the correct choice is Yes: the equation in the image is a proper and standard rendering of ∫_0^1 x^2 dx as the limit of the Riemann sum (1/n) Σ (k/n)^2 and evaluates to 1/3, so it renders correctly. Why "No" is incorrect: there is no omission or misrendering of the mathematical content—the sum, the 1/n factor, and the limiting process are all present and consistent with the integral. Therefore the correct answer is Yes, which matches the provided answer.

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 单项选择题

类似问题

Which of the following is the upper Riemann sum \(U_{5}\) over 5 subintervals for the integral \[ \int _{2}^{3} \sqrt {4 + x^{2}} \, dx, \] correct to three decimal places?

Which of the following is the upper Riemann sum 𝑈 3 over 3 subintervals for the integral ∫ 3 5 4 + 𝑥 2 𝑑 𝑥 , correct to three decimal places?

Which of the following is the upper Riemann sum 𝑈 5 over 5 subintervals for the integral ∫ 2 3 4 + 𝑥 2 𝑑 𝑥 , correct to three decimal places?

Which of the following is the lower Riemann sum L5 over 5 subintervals for the integral ∫ 3 0 √ 7+x4 dx, correct to three decimal places?

Which of the following is the lower Riemann sum 𝐿 4 over 4 subintervals for the integral ∫ 2 5 1 + 𝑥 2 𝑑 𝑥 , correct to three decimal places?

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.

Express the following integral as a limit of Riemann sums:

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单