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.