Area under Curve/Examples/(x - 1) (x - 2)

Example of Use of Area under Curve
The area between the $x$-axis and the curve $y = \paren {x - 1} \paren {x - 2}$ is $\dfrac 1 6$.

Proof
Let $\AA$ be the area in question.

The curve $y = \paren {x - 1} \paren {x - 2}$ intercepts the $x$-axis where $y = 0$.

That is, where $x - 1 = 0$ and $x - 2 = 0$, which is $\tuple {1, 0}$ and $\tuple {2, 0}$.


 * Area-under-Curve-(x-1)(x-2).png

Thus from Area under Curve we need to evaluate the definite integral:
 * $\AA = \ds \int_1^2 \paren {x - 1} \paren {x - 2} \rd x$

Between those limits, $y$ is negative.

Hence we expect a negative result, which must then be made positive.

So: