Question

The RMS value of \(x^2\) in \([0,1]\) is

• A. \(\frac{1}{\sqrt5}\)
• B. \(\frac{1}{5}\)
• C. \(\frac{1}{\sqrt3}\)
• D. \(\frac{1}{3}\)

Hint:

RMS value means square the function, take its average over the interval, and then take the square root.

Answer 1

The Correct Option is A

We need to find the RMS value of the function: \[ f(x)=x^2 \] in the interval \[ [0,1]. \] The RMS value of a function \(f(x)\) in the interval \([a,b]\) is: \[ \text{RMS}=\sqrt{\frac{1}{b-a}\int_a^b [f(x)]^2\,dx}. \] Here, \[ a=0,\qquad b=1. \] So, \[ b-a=1-0=1. \] Also, \[ f(x)=x^2. \] Therefore, \[ [f(x)]^2=(x^2)^2=x^4. \] Now apply the RMS formula: \[ \text{RMS}=\sqrt{\frac{1}{1}\int_0^1 x^4\,dx}. \] \[ \text{RMS}=\sqrt{\int_0^1 x^4\,dx}. \] Now integrate: \[ \int x^4\,dx=\frac{x^5}{5}. \] Therefore, \[ \int_0^1 x^4\,dx=\left[\frac{x^5}{5}\right]_0^1. \] \[ =\frac{1^5}{5}-\frac{0^5}{5}. \] \[ =\frac{1}{5}. \] Hence, \[ \text{RMS}=\sqrt{\frac{1}{5}}. \] \[ \text{RMS}=\frac{1}{\sqrt5}. \] Therefore, the RMS value is \[ \frac{1}{\sqrt5}. \]