Answer
$\approx3.627$
Work Step by Step
The formula for the arc length is
$s=\int_{a}^b \sqrt{1 + (y')^2} dy$
Differentiate $y = \frac{1}{2}(e^x + e^{−x})$ with respect to x.
$y' = \frac{1}{2}(e^x - e^{−x})$
Substitute the value of y' in $1 + (y')^2$
$1+(y')^2=1+[\frac{1}{2}(e^x - e^{−x})]^2$
$=\frac{1}{4}(e^x + e^{−x})^2$
Substitute the value of $1 + (y')^2$ in the formula for the arc length $s=\int_{a}^b \sqrt{1 + (y')^2} dy$
Solve for the distance s
$s=\int_{0}^2\sqrt{\frac{1}{4}(e^x - e^{-x})}^{2} dx$
$=\int_{0}^2 \frac{1}{2}(e^x - e^{-x})dx$
$=\frac{1}{2}[e^x - e^{-x}]_0^2\ $
$= \frac{1}{2}(e^2 - e^{-2})$
$\approx3.627$
