Question

Sketch the curves and identify the region bounded by $x=\frac{1}{2},x=2,y=\ln x$ and $y=2^x$. Find the area of this region.

Answer 1

The given curves are
$x=\frac{1}{2}$
$x=2$
$y={\log }_{e}x$
$y=2^x$
Required area $=ABCDA$
$={\int }_{1/2}^2(2^x-\log x)dx$
$={[\frac{2^x}{\log 2}-(x\log x-x\left)\right]}_{1/2}^2$
$=(\frac{4}{\log 2}-2\log 2+2)-(\frac{\sqrt{2}}{\log 2}-\frac{1}{2}\log \frac{1}{2}+\frac{1}{2})$
$=(\frac{4-\sqrt{2}}{\log 2}-\frac{5}{2}\log 2+\frac{3}{2})sq.$ units.