Area Question 51
Question 51
- Find the area of the region bounded by the $X$-axis and the curves defined by $y=\tan x,-\frac{\pi}{3} \leq x \leq \frac{\pi}{3}$ and $y=\cot x, \frac{\pi}{6} \leq x \leq \frac{\pi}{3}$.
$(1984,4 M)$
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Solution:
- Given, $y=\tan x,-\frac{\pi}{3} \leq x \leq \frac{\pi}{3}$ $\cot x, \quad \frac{\pi}{6} \leq x \leq \frac{\pi}{2}$
which could be plotted as $Y$-axis.
$\therefore$ Required area $=\int_{0}^{\pi / 4}(\tan x) d x+\int_{\pi / 4}^{\pi / 3}(\cot x) d x$
$=[-\log |\cos x|]{0}^{\pi / 4}+[\log \sin x]{\pi / 4}^{\pi / 3}$
$=-\log \frac{1}{\sqrt{2}}-0+\log \frac{\sqrt{3}}{2}-\log \frac{1}{\sqrt{2}}$
$=\log \frac{\sqrt{3}}{2}-2 \log \frac{1}{\sqrt{2}}$
$=\log \frac{\sqrt{3}}{2}-\log \frac{1}{2}=\frac{1}{2} \log _{e} 3$ sq units
