# What is the net area between f(x) = 4/x  and the x-axis over x in [1, 2 ]?

May 29, 2018

$4 \ln 2 \approx 2.7725887 \ldots$

#### Explanation:

We seek the net area between $f \left(x\right) = \frac{4}{x}$ and the $x$-axis for $x \in \left[1 , 2\right]$.

We first note that $f \left(x\right)$ has a disvontunitry at $x = 0$ and that this discontinuity is outside the desired range, and so is of no concern. Further noting that $f \left(x\right)$ is continuous and positive over the desired range, the net area is given by:

$A = {\int}_{1}^{2} \setminus \frac{4}{x} \setminus \mathrm{dx}$
$\setminus \setminus = 4 {\left[\setminus \ln | x | \setminus\right]}_{1}^{2}$

$\setminus \setminus = 4 \left(\ln 2 - \ln 1\right)$

$\setminus \setminus = 4 \ln 2$