# What is the arc length of the curve given by f(x)=1+cosx in the interval x in [0,2pi]?

##### 1 Answer
Apr 13, 2018

The arc length is most nearly $7.64$ units.

#### Explanation:

Recall that arc length is given by $A = {\int}_{a}^{b} \sqrt{1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{2}} \mathrm{dx}$.

The derivative of $f ' \left(x\right)$ is $f ' \left(x\right) = - \sin x$.

$A = {\int}_{0}^{2 \pi} \sqrt{1 + {\left(- \sin x\right)}^{2}} \mathrm{dx}$

$A = {\int}_{0}^{2 \pi} \sqrt{1 + {\sin}^{2} x}$

An approximation using a calculator gives $A = 7.64$.

Hopefully this helps!