Page:Calculus Made Easy.pdf/228

integrate $y\cdot dx$ for the particular case, when we know what the value of $y$ is as a function of $x$.

For instance, if you were told that for the particular curve in question $y=b+ax^{2}$, no doubt you could put that value into the expression and say: then I must find $\int (b+ax^{2})\,dx$.

That is all very well; but a little thought will show you that something more must be done. Because the area we are trying to find is not the area under the whole length of the curve, but only the area limited on the left by $PM$, and on the right by $QN$, it follows that we must do something to define our area between those ‘limits’.

This introduces us to a new notion, namely that of integrating between limits. We suppose $x$ to vary, and for the present purpose we do not require any value of $x$ below $x_{1}$ (that is $OM$), nor any value of $x$ above $x_{2}$ (that is $ON$). When an integral is to be thus defined between two limits, we call the lower of the two values the inferior limit, and the upper value the superior limit. Any integral so limited we designate as a definite integral, by way of distinguishing it from a general integral to which no limits are assigned.

In the symbols which give instructions to integrate, the limits are marked by putting them at the top and bottom respectively of the sign of integration. Thus the instruction

$\int _{x=x_{1}}^{x=x_{2}}y\cdot dx$