\begin{definition}[Definition:Integral Curve]
Let $\map f {x, y}$ be a continuous real function within a rectangular region $R$ of the Cartesian plane.
Consider the first order ODE:
:$(1): \quad \dfrac {\d y} {\d x} = \map f {x, y}$
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Let $P_0 = \tuple {x_0, y_0}$ be a point in $R$.
The number:
:$\paren {\dfrac {\d y} {\d x} }_{P_0} = \map f {x_0, y_0}$
determines the slope of a straight line passing through $P_0$.
Let $P_1 = \tuple {x_1, y_1}$ be a point in $R$ close to $P_0$.
Then:
:$\paren {\dfrac {\d y} {\d x} }_{P_1} = \map f {x_1, y_1}$
determines the slope of a straight line passing through $P_1$.
Let $P_2 = \tuple {x_2, y_2}$ be a point in $R$ close to $P_1$.
Then:
:$\paren {\dfrac {\d y} {\d x} }_{P_2} = \map f {x_2, y_2}$
determines the slope of a straight line passing through $P_2$.
Continuing this process, we obtain a curve made of a sequence of straight line segments.
As successive points $P_0, P_1, P_2, \ldots$ are taken closer and closer to each other, the sequence of straight line segments $P_0 P_1 P_2 \ldots$ approaches a smooth curve $C$ passing through an initial point $P_0$.
By construction, for each point $\tuple {x, y}$ on $C$, the slope of $C$ at $\tuple {x, y}$ is given by $\map f {x, y}$.
Hence this curve is a solution to $(1)$.
Starting from a different point, a different curve is obtained.
Thus the general solution to $(1)$ takes the form of a set of curves.
This set of curves are referred to collectively as '''integral curves'''.
\end{definition}