Area Under a Curve using Integration

If $f(x)$ is positive and continuous on the interval $1 \le x \le 4$, then the area bounded by $y=f(x)$, the $x$-axis, and the vertical lines $x=1$ and $x=4$ is given by: $$\displaystyle A=\int_{1}^{4}{f(x)}dx$$ Area Under a Curve

Example 1

Find the area of the region bounded by $y=2x$, $x$-axis, $x=1$ and $x=5$.

Example 2

Find the area enclosed by $y=\sin{2x}$, $x=0$, $x=\displaystyle \dfrac{\pi}{2}$ and the $x$-axis.

Example 3

Find the area enclosed by $y=\dfrac{1}{x}$, $x=1$, $x=3$ and the $x$-axis, correcting to two significant figures.