Page:Calculus Made Easy.pdf/243

Examples.

(1) To find the quadratic mean of the function $y=ax$ (Fig. 63).

Here the integral is $\int _{0}^{l}a^{2}x^{2}\,dx$, which is ${\frac {1}{3}}a^{2}l^{3}$.

Fig. 63.

Dividing by $l$ and taking the square root, we have

quadratic mean $={\frac {1}{\sqrt {3}}}\,al$.

Here the arithmetical mean is ${\tfrac {1}{2}}al$; and the ratio of quadratic to arithmetical mean (this ratio is called the form-factor) is ${\dfrac {2}{\sqrt {3}}}=1.155$.

(2) To find the quadratic mean of the function $y=x^{a}$.

The integral is $\int _{x=0}^{x=l}x^{2a}\,dx$, that is ${\dfrac {l^{2a+1}}{2a+1}}$.

Hence quadratic mean $={\sqrt[{2}]{\dfrac {l^{2a}}{2a+1}}}$.

(3) To find the quadratic mean of the function $y=a^{\frac {x}{2}}$ The integral is $\int _{x=0}^{x=l}(a^{\frac {x}{2}})^{2}\,dx$, that is $\int _{x=0}^{x=l}a^{x}\,dx$,