True or False? If f is continuous on [a,b], then

∫
b
a
f(x)dx=lim
n→∞
n
∑

i=1f(xi)Δx where Δx=
b−a

n
and xi=a+iΔx .

Question

True or False? If f is continuous on [a,b], then

∫
b
a
f(x)dx=lim
n→∞
n
∑

i=1f(xi)Δx  where Δx=
b−a

n
  and xi=a+iΔx .

BlackTom AI · Accepted Answer

Consider the statement about the Riemann integral and uniform partitions. 
Option 1 (True): The definite integral ∫_a^b f(x) dx equals the limit of the Riemann sums as the mesh of the partition goes to zero. When f is continuous on [a,b], the function is integrable, and for the regular partition with Δx = (b−a)/n and sample points xi = a + iΔx, the Riemann sum ∑_{i=1}^n f(xi) Δx converges to the integral as n → ∞. This is precisely the standard definition/theorem connecting Riemann sums to the definite integral for continuous functions.

Option 2 (False): The statement would be false if it claimed the limit fails to exist or if continuity were not sufficient. However, continuity on [a,b] guarantees integrability and the limit of the equal-width Riemann sums exists and equals the integral. The only potential caveat would be if the sample points xi were chosen in a way that did not reflect a proper Riemann sum, but here they are the right-endpoints of a uniform partition, which still yields convergence to ∫_a^b f(x) dx for continuous f.

In summary, the continuity of f on the closed interval ensures the Riemann sums with uniform partition converge to the definite integral, making the True option consistent with the fundamental relationship between Riemann sums and the integral. The False option misstates this standard result by implying a failure or limitation that does not exist for continuous functions on [a,b].

MATH_1225_17243_202501 5.2 The Definite Integral

True or False? If f is continuous on [a,b], then ∫ b a f(x)dx=lim n→∞ n ∑ i=1f(xi)Δx where Δx= b−a n and xi=a+iΔx .

查看解析

登录即可查看完整答案

类似问题

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

People born without creativity never can develop it

A product has a selling price of $20, a contribution margin ratio of 40% and fixed cost of $120,000. To make a profit of $30,000. The number of units that must be sold is: Type the number without $ and a comma. Eg: 20000

Which of the following statement regarding cost is correct:

Under the assumptions used in cost-volume-profit analysis, as the activity increases:

The CVP and break-even analysis are supported by a range of important assumptions. Which of the following is not an assumption of the CVP:

The break-even point is that at which:

更多留学生实用工具

考试浏览器助手

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