Step 1: Understanding the Concept:
The Root Mean Square (RMS) value of a function $f(x)$ over an interval $[a, b]$ is a measure of the magnitude of the function. It is the square root of the mean (average) of the square of the function.
Step 2: Key Formula or Approach:
The formula for the RMS value of a function $f(x)$ on the interval $[a, b]$ is:
\[ \text{RMS} = \sqrt{\frac{1}{b-a} \int_a^b [f(x)]^2 \,dx} \]
Here, $f(x) = x^2$, $a=0$, and $b=1$.
Step 3: Detailed Explanation:
1. Square the function:
\[ [f(x)]^2 = (x^2)^2 = x^4 \]
2. Mean (average) of the square over the interval:
This is calculated by integrating the squared function and dividing by the length of the interval.
\[ \text{Mean of square} = \frac{1}{1-0} \int_0^1 x^4 \,dx = 1 \cdot \left[ \frac{x^5}{5} \right]_0^1 \]
\[ = \left( \frac{1^5}{5} - \frac{0^5}{5} \right) = \frac{1}{5} \]
3. Root of the mean:
Take the square root of the result from the previous step.
\[ \text{RMS} = \sqrt{\frac{1}{5}} = \frac{1}{\sqrt{5}} \]
Step 4: Final Answer:
The RMS value of $x^2$ in the interval [0, 1] is $\frac{1}{\sqrt{5}}$. Therefore, option (A) is correct.