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.

Fig. 50.

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