If $f(x)=2\ln (\sqrt{e^x})$, what is the area bounded by f(x) for the interval [0, 2] on the x-axis?

Answer :

Solution :

$y=f(x)=2\ln (\sqrt{e^x})$

The area bounded by the curve y = f(x) and in the interval where x ∈ (0, 2) is given by :

$A={\int }_0^{2}2\ln (\sqrt{e^x})dx$

$A={\int }_0^{2}2\ln (e^{x/2})dx$

$A={\int }_0^{2}2\times \frac{x}{2}dx$

$A={\left[\frac{x^2}{2}\right]}_0^2=[2-0]=2$