Area Between Two Functions



If two functions $f(x)$ and $g(x)$ intersect at $x=1$ and $x=3$, and $f(x) \ge g(x)$ for all $1 \le x \le 3$, then the area of the shaded region between their points of intersection is given by:
$$ \begin{align} \displaystyle
A &= \int_{1}^{3}{f(x)}dx – \int_{1}^{3}{g(x)}dx \\
&= \int_{1}^{3}{\Big[f(x)-g(x)\Big]}dx
\end{align} $$

Example 1

Find the area bounded by the $x$-axis and $y=x^2-4x+3$.

Example 2

Find the area bounded by the $y=x+2$ and $y=x^2+x-2$.

Example 3

Find the area bounded by the $x$-axis and $y=x^3-x^2-2x$.






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