Question

Find the root mean square value of f(x).=x² in (0,l)

Answer 1

Answer:

find square root mean square value of function f(x)=x² in intervals (0,l)

Answer 2

Given,

f(x) = x²

To find,

The RMS value of f(x) in (0,1).

Solution,

The root mean square value of f(x)=x² in (0,1) is $\sqrt{\frac{1}{5} }$.

We can simply solve the mathematical problem by the following procedure.

We know that,

RMS value = $\sqrt{\frac{\int\limits^1_0 {y^2} \, dx }{\int\limits^1_0 {} \, dx } }$

Putting the value of 'x' in place of 'y'.

                  = $\sqrt{\frac{\int\limits^1_0 {x^4} \, dx }{\int\limits^1_0 {} \, dx } }$

Following the rules of integration.

                  = $\sqrt{\frac{\frac{x^5}{5} }{x}$

                  = $\sqrt{\frac{\frac{1}{5} }{1}}$

                  = $\sqrt{\frac{1}{5} }$

Thus, the root mean square value of f(x)=x² is $\sqrt{\frac{1}{5} }$.