Question

The area (in sq. units) of the region bounded by \[ x=0,\quad x=\frac{\pi}{2}, \] \[ y=0,\quad y=\cos x,\quad \text{and}\quad y=\tan x \] is

• A. \[ \frac{\sqrt5-1}{2} + \frac12 \log\left(\frac{\sqrt5-1}{2}\right) \]
• B. \[ \frac{3-\sqrt5}{2} + \log\left(\frac{\sqrt5-1}{2}\right) \]
• C. \[ \frac{\sqrt5-1}{2} - \log\left(\frac{\sqrt5-1}{2}\right) \]
• D. \[ \frac{3-\sqrt5}{2} + \frac12 \log\left(\frac{\sqrt5+1}{2}\right) \]

Hint:

When multiple curves bound a region, first find their intersection points. Then split the area into intervals where the upper boundary changes and integrate accordingly.

Answer 1

The Correct Option is D