Question

Suppose $f:[a,b]\to\mathbb{R}$ is Riemann integrable. Prove that $\int_{a}^{b}f=\lim_{n\to\infty} \frac{b-a}{n}\sum_{j=1}^{n} f(a+\frac{j(b-a)}{n}).$

score 451 · Answer 1 · 2023-02-13 10:40:01

Suppose $f:[a,b]\to\mathbb{R}$ is Riemann integrable. Prove that $\int_{a}^{b}f=\lim_{n\to\infty} \frac{b-a}{n}\sum_{j=1}^{n} f(a+\frac{j(b-a)}{n}).$

score 1.6k · Answer 2 · 2023-02-13 11:05:46

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List Question

Suppose $f:[a,b]\to\mathbb{R}$ is Riemann integrable. Prove that $\int_{a}^{b}f=\lim_{n\to\infty} \frac{b-a}{n}\sum_{j=1}^{n} f(a+\frac{j(b-a)}{n}).$

Why do we need Lebesgue Integration and what are the limits to Riemann Integrals?

I wonder why Michael Spivak assumed $f$ is integrable on $[a,b]$ "Calculus 4th Edition" by Michael Spivak "Measure, Integration & Real Analysis" Axler

Does (sin(1/x))^2 have an improper integral in between -inf and inf

Conditions Necessary to Determine Outer Measure Equals the Measure of a Set

Proving that $\sum_{n=1}^{\infty} \frac{\sin(nx)}{n^\alpha}$ is not the Fourier Series of a Riemann Integrable function

Pythagorean Integral Inequality

Tried solving this using 2nd fundamental thm of calculus... But

A problem from Applied Asymptotic Analysis by Peter D. Miller (Exercise 1.8)

Proving that the image of a Riemann sums of a function with the intermediate value property(Darboux function) is an interval

Does this process for building partitions lead to convergence to Riemann integral?

Riemann Integral of finite rational activation function.

Are two Riemann integrable functions, Riemann stieltjes integrable with respect ro each other?

A Riemann Sum clarification

Does being equal to a riemann integrable function almost everywhere imply riemann integrability for a bounded function?

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