Page:Calculus Made Easy.pdf/243

 Examples.

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

Here the integral is $$\int^l_0 a^2 x^2\, dx$$, which is $$\frac{1}{3} a^2 l^3$$.



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

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

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