Page:Calculus Made Easy.pdf/226

CHAPTER XIX.

ON FINDING AREAS BY INTEGRATING.

One use of the integral calculus is to enable us to ascertain the values of areas bounded by curves.

Let us try to get at the subject bit by bit.

Fig. 52.

Let $AB$ (Fig. 52) be a curve, the equation to which is known. That is, $y$ in this curve is some known function of $x$. Think of a piece of the curve from the point $P$ to the point $Q$.

Let a perpendicular $PM$ be dropped from $P$, and another $QN$ from the point $Q$. Then call $OM=x_{1}$ and $ON=x_{2}$, and the ordinates $PM=y_{1}$ and $QN=y_{2}$. We have thus marked out the area $PQNM$ that lies