Question

Area of the region bounded by the curve $y = tan x$, the x- axis and the line $x = \frac{\pi}{3}$ is

• A. $\log \frac{1}{2}$
• B. 0
• C. log 2
• D. - log 2

Answer 1

The Correct Option is C

The region lies between $x = 0$ and $x = \frac{\pi}{3}$ 
We can find the area by integrating the absolute value of the function 
$y = tan x$  within this interval. 
Thus, the area (A) is given by: 
$A = \int_{0}^{\pi/3} | \tan x | \, dx$
To evaluate this integral, we need to break it up into two parts due to the nature of the tangent function. 
$A = \int_{0}^{\pi/3} \tan x \, dx - \int_{0}^{\pi/3} (-\tan x) \, dx$

Simplifying this expression, we have: 
$A = \int_{0}^{\pi/3} \tan x \, dx + \int_{0}^{\pi/3} \tan x \, dx$
Combining the integrals, we get: 
$A = 2 \int_{0}^{\pi/3} \tan x \, dx$

Using the integral property, we have: 
$A = 2 \left[\log | \sec x | \right]_{0}^{\pi/3} A$
$=$ $2 \left[\log(|\sec(\frac{\pi}{3})|) - \log(|\sec 0|)\right] A$ 
$= 2 [log (2) - log (1)] A$
$= 2 log (\frac{2}{1}) A$
$= 2 log (2)$
Therefore, the area of the region bounded by the curve $y = tan x$, the x-axis, and the line $x = \frac{\pi}{3} \text{ is } 2 \log(2)$
 which corresponds to option (C) log 2.