Page:Calculus Made Easy.pdf/238

 of the base $$ON$$. But before we can find the area we must ascertain the length of the base, so as to know up to what limit we are to integrate. At $$N$$ the ordinate $$y$$ has zero value; therefore, we must look at the equation and see what value of $$x$$ will make $$y=0$$. Now, clearly, if $$x$$ is $$0$$, $$y$$ will also be $$0$$, the curve passing through the origin $$O$$; but also, if $$x=1$$, $$y=0$$; so that $$x=1$$ gives us the position of the point $$N$$.

Then the area wanted is

But the base length is $$1$$.

Therefore, the average ordinate of the curve $$= \tfrac{1}{6}$$.

[N.B.–It will be a pretty and simple exercise in maxima and minima to find by differentiation what is the height of the maximum ordinate. It must be greater than the average.]

The mean ordinate of any curve, over a range from $$x=0$$ to $$x=x_1$$, is given by the expression,

One can also find in the same way the surface area of a solid of revolution.