Page:Calculus Made Easy.pdf/209

 for a perfect curve we ought to take each $$dx$$ and its corresponding $$dy$$ infinitesimally small, and infinitely numerous.



Then, how much ought the value of any $$y$$ to be? Clearly, at any point $$P$$ of the curve, the value of $$y$$ will be the sum of all the little $$dy$$’s from $$0$$ up to that level, that is to say, $$\int dy = y$$. And as each $$dy$$ is equal to $$\int \tfrac{1}{5}x \cdot dx$$, it follows that the whole $$y$$ will be equal to the sum of all such bits as $$\tfrac{1}{5}x \cdot dx$$, or, as we should write it, $$\int \tfrac{1}{5}x \cdot dx$$.

Now if $$x$$ had been constant, $$\int \tfrac{1}{5}x \cdot dx$$ would have been the same as $$\tfrac{1}{5} x \int dx$$, or $$\tfrac{1}{5}x^2$$. But $$x$$ began by being $$0$$, and increases to the particular value of $$x$$ at the point $$P$$, so that its average value from $$0$$ to that point is $$\tfrac{1}{2}x$$. Hence $$\int \tfrac{1}{5} x\, dx = \tfrac{1}{10} x^2$$; or $$y=\tfrac{1}{10}x^2$$.

But, as in the previous case, this requires the addition of an undetermined constant $$C$$, because we have not