Root Mean Square Value Of A Function

Suppose y = f(x) is the equation of a curve. Then the root mean square (rms) value of the function between x = x_1 ad x = x_2 will be

Now I’ll give some examples of that.

EXAMPLE 1

According to Stroud and Booth (2013)*, “Find the rms value of $i = \cos x + \sin x$ over the range x = 0 to $\cfrac{3 \pi}{4}$ .”

SOLUTION

Now here the given function is $i = \cos x + \sin x$ . And I have to find its rms value over the range x = 0 to $\cfrac{3 \pi}{4}$ . Thus it will be

And that means

So this gives

Now I’ll integrate it using the standard formulas in integration. And that means

Next, I’ll substitute the limits to get

Then I’ll simplify it. And that gives

Therefore the rms value of the function is

Hence I can conclude that this is the answer to the given example.

Now I’ll give another example.

EXAMPLE 2

According to Stroud and Booth (2013)*, “Calculate the rms value of $i = 20 + 100 \sin 100 \pi t$ between t = 0 and t = 1/50 .”

SOLUTION

Now here the given function is $i = 20 + 100 \sin 100 \pi t$ . And I have to get its rms values between t = 0 and t = 1/50 . Thus it will be

And that means

So this gives

Then I’ll integrate it to get

Next, I’ll substitute the limits to get

Then I’ll simplify it. So that gives

And this is because $\cos 0 = 1, \cos 2 \pi = 1, \sin 4 \pi = 0$ .

Now I’ll simplify it a bit more to get the value of $\text{rms}^2$ as

So I can rewrite it as

And that means

Therefore the rms value of the function will be

Hence I can conclude that this is the answer to the given example.

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