Question

Let $f:\left[0, \infty \right]\ \to \mathbb{R}$ be a nonnegative function such that $\int_{0}^{\infty} f(x) \,dx <\infty$, that $f$ is Riemann integrable on any finite interval $\left[0,R\right]$ and that $\lim_{R\to\infty}\int_{0}^{R} f(x) \,dx $ exists and is finite.Prove that the limit $\lim_{R \to \infty}\frac{1}{R}\int_{0}^{R} xf(x) \,dx $ exists and determine its value.

Name: let fleft0 infty right to mathbbr be a nonnegative function such that int0infty fx dx infty that f is riemann integrable on any finite interval left0rright and that limrtoinftyint0r fx dx exists an
Uploaded: 2023-06-03T10:05:41-08:00
Duration: 2 min 8 s
Channel: Vincenzo Zaccaro
Description: let fleft0 infty right to mathbbr be a nonnegative function such that int0infty fx dx infty that f is riemann integrable on any finite interval left0rright and that limrtoinftyint0r fx dx exists an

          Let  $f:\left[0, \infty \right]\ \to \mathbb{R}$   be a nonnegative function such that   $\int_{0}^{\infty} f(x) \,dx <\infty$,  that  $f$  is Riemann integrable on any finite interval  $\left[0,R\right]$  and that $\lim_{R\to\infty}\int_{0}^{R} f(x) \,dx $ exists and is finite.Prove that the limit $\lim_{R \to \infty}\frac{1}{R}\int_{0}^{R} xf(x) \,dx $ exists and determine its value.

Added by Tutku D.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Vincenzo Zaccaro

06/03/2023

Step 1

This means that the function $f(x)$ must approach 0 as $x$ approaches infinity, otherwise the integral would not converge. Now, let's consider the limit we want to find: $\lim_{R \to \infty}\frac{1}{R}\int_{0}^{R} xf(x) \,dx$. We can rewrite this limit as Show more…

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Let $f:\left[0, \infty \right]\ \to \mathbb{R}$ be a nonnegative function such that $\int_{0}^{\infty} f(x) \,dx <\infty$, that $f$ is Riemann integrable on any finite interval $\left[0,R\right]$ and that $\lim_{R\to\infty}\int_{0}^{R} f(x) \,dx $ exists and is finite.Prove that the limit $\lim_{R \to \infty}\frac{1}{R}\int_{0}^{R} xf(x) \,dx $ exists and determine its value.