# What is the arc length of f(x)=2x-1 on x in [0,3]?

Dec 7, 2015

$3 \sqrt{5}$

#### Explanation:

Arc length can be found using the integral;

$L = {\int}_{a}^{b} \sqrt{1 + f ' {\left(x\right)}^{2}} \mathrm{dx}$

You can find the derivation of this formula here . To start, we need to know the derivative, $f ' \left(x\right)$, which can be found using the power rule.

$\frac{d}{\mathrm{dx}} \left(2 x - 1\right) = 2$

Now we can set up the integral.

$L = {\int}_{0}^{3} \sqrt{1 + {2}^{2}} \mathrm{dx}$

$= {\int}_{0}^{3} \sqrt{5} \mathrm{dx}$

$= \sqrt{5} {\int}_{0}^{3} \mathrm{dx}$

$= \sqrt{5} {\left[x\right]}_{0}^{3}$

$= 3 \sqrt{5}$