Question
What is the area bounded by $$y = \tan \,x,\,y = 0$$ and $$x = \frac{\pi }{4}\,?$$
A.
$$\ln\,2$$ square units
B.
$$\frac{{\ln\,2}}{2}$$ square units
C.
$$2\left( {\ln} \right)2$$ square units
D.
None of these
Answer :
$$\frac{{\ln\,2}}{2}$$ square units
Solution :
Required area
$$\eqalign{
& = \int_0^{\frac{\pi }{4}} {\tan \,x\,dx} \cr
& = \ln \left| {\sec \,1} \right|_0^{\frac{\pi }{4}} \cr
& = \ln \sqrt 2 \cr
& = \frac{{\ln \,2}}{2} \cr} $$