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FINDING AREAS BY INTEGRATING

211

All integration between limits requires the difference between two values to be thus found. Also note that, in making the subtraction the added constant $C$ has disappeared.

Examples

(1) To familiarize ourselves with the process, let us take a case of which we know the answer beforehand. Let us find the area of the triangle (Fig. 53), which

Fig. 53.

has base $x=12$ and height $y=4$ . We know beforehand, from obvious mensuration, that the answer will come $24$ .

Now, here we have as the “curve” a sloping line for which the equation is

$y={\frac {x}{3}}$ .

The area in question will be

$\int _{x=0}^{x=12}y\cdot dx=\int _{x=0}^{x=12}{\frac {x}{3}}\cdot dx$ .

Integrating ${\dfrac {x}{3}}\,dx$ (p. 194), and putting down the

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