Let ƒ be an integrable function on [a,b], and $s _ { n }$ and $s _ { n }$ be the lower and upper sum of a partition of [a,b], respectively. Which of the following is always true?

A) $s _ { n } = S _ { n }$
B) $\lim _ { n \rightarrow \infty } s _ { n } = \lim _ { n \rightarrow \infty } S _ { n }$
C) $\int _ { a } ^ { b } f ( x ) d x \neq 0$
D)ƒ is continuous on [a,b].
E) $\int _ { a } ^ { b } f ( x ) d x \geq 0$

Question

Let ƒ be an integrable function on [a,b], and $s _ { n }$ and $s _ { n }$ be the lower and upper sum of a partition of [a,b], respectively. Which of the following is always true?

A) $s _ { n } = S _ { n }$
B) $\lim _ { n \rightarrow \infty } s _ { n } = \lim _ { n \rightarrow \infty } S _ { n }$
C) $\int _ { a } ^ { b } f ( x ) d x \neq 0$
D)ƒ is continuous on [a,b].
E) $\int _ { a } ^ { b } f ( x ) d x \geq 0$

Quizplus · Accepted Answer

The limit of the lower sums ($s_n$) and the limit of the upper sums ($S_n$) as $n$ approaches infinity are equal for an integrable function on [a,b], and this common limit is the definite integral of the function over [a,b]. This is a fundamental property of Riemann integrals.

Let ƒ Be an Integrable Function on [A,b], And sns _ { n }sn​

Let ƒ Be an Integrable Function on [A,b], And $s _ { n }$