Q. The area of the region bounded by the curves $ y=ex,\text{ y}=lo{{g}_{e}}x $ and lines $ x=1,\text{ }x=2 $ is
Rajasthan PETRajasthan PET 2009
Solution:
Required area, $ A=\int_{1}^{2}{({{e}^{x}}-{{\log }_{e}}x)}\,dx $
$ =[{{e}^{x}}]_{1}^{2}-[x{{\log }_{e}}x-\int{1\,dx}]_{1}^{2} $
$ ={{e}^{2}}-e-[x{{\log }_{e}}x-x]_{1}^{2} $
$ ={{e}^{2}}-e-[2{{\log }_{e}}2-2(0-1)] $
$ ={{e}^{2}}-e-2{{\log }_{e}}2+1 $
