Question

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

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

Hint:

Remember the three steps for RMS: Square the function, find the Mean (average) of the result over the interval, and then take the square Root. Following the steps in reverse order (S-M-R) helps in remembering the process.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to calculate the Root Mean Square (RMS) value of the function \(f(x) = x^2\) over the interval \([0, 1]\).

Step 2: Key Formula or Approach:
The RMS value of a function \(f(x)\) over the interval \([a, b]\) is given by the formula:
\[ \text{RMS} = \sqrt{\frac{1}{b-a} \int_a^b [f(x)]^2 \,dx} \]

Step 3: Detailed Explanation:
Here, the function is \(f(x) = x^2\) and the interval is \([a, b] = [0, 1]\).
First, we find the square of the function:
\[ [f(x)]^2 = (x^2)^2 = x^4 \]
Next, we calculate the mean square value, which is the average of the squared function over the interval.
\[ \text{Mean Square} = \frac{1}{1-0} \int_0^1 x^4 \,dx = \int_0^1 x^4 \,dx \]
Evaluate the integral:
\[ \int_0^1 x^4 \,dx = \left[ \frac{x^5}{5} \right]_0^1 = \frac{1^5}{5} - \frac{0^5}{5} = \frac{1}{5} \]
Finally, the RMS value is the square root of the mean square value:
\[ \text{RMS} = \sqrt{\frac{1}{5}} = \frac{1}{\sqrt{5}} \]

Step 4: Final Answer:
The RMS value of \(x^2\) in the interval [0, 1] is \(\frac{1}{\sqrt{5}}\).