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 .

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