How do you find the arc length of the curve y=lnx from [1,5]?

Sep 12, 2015

The arc length is defined by the following formula: $L = {\int}_{1}^{5} \sqrt{\frac{{x}^{2} + 1}{x}} \mathrm{dx}$

Explanation:

To find the arc length of a curve, we use the formula
$L = {\int}_{a}^{b} \sqrt{1 + {\left[f ' \left(x\right)\right]}^{2}} \mathrm{dx}$

In our case, $f \left(x\right) = \ln x$, $a = 1$, and $b = 5$.
The derivative of $f \left(x\right) = \ln x$ is $f ' \left(x\right) = \frac{1}{x}$, so we get:

$L = {\int}_{1}^{5} \sqrt{1 + {\left[\frac{1}{x}\right]}^{2}} \mathrm{dx}$
$L = {\int}_{1}^{5} \sqrt{1 + \frac{1}{{x}^{2}}} \mathrm{dx}$
$L = {\int}_{1}^{5} \sqrt{\frac{{x}^{2} + 1}{x}} \mathrm{dx}$