Page:Calculus Made Easy.pdf/243

Examples.

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

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

Dividing by ${\displaystyle l}$ and taking the square root, we have

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

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

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

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