Question

The value of $\underset{\pi /6}{\overset{5\pi /6}{\int }}\sqrt{4-4{\sin }^{2}t}dt$ is

• A. 0
• B. 2
• C. 1
• D. none of these.

Answer 1

Answer: (B)

$I=\underset{\pi 6}{\overset{5\pi /6}{\int }}\sqrt{4-4si{n}^{2}t}dt=2\underset{\pi /6}{\overset{5\pi /6}{\int }}\left|cost\right|dt$
$=2\underset{\pi /6}{\overset{\pi /2}{\int }}\left|cost\right|dt+2\underset{\pi /2}{\overset{5\pi /6}{\int }}\left|cost\right|dt$
$=2\underset{\pi /6}{\overset{\pi /2}{\int }}costdt+2\underset{\pi /2}{\overset{5\pi /6}{\int }}-costdt$
$=2\cdot {\left|sint\right|}_{\pi /6}^{\pi /2}-2{\left|sint\right|}_{\pi /2}^{5\pi /6}$
$=2\left[sin\frac{\pi }{2}-sin\frac{\pi }{6}\right]-\left[sin\frac{5\pi }{6}-sin\frac{\pi }{2}\right]$
$=2\left(1-\frac{1}{2}\right)-2\left(\frac{1}{2}-1\right)$
$=2\left(\frac{1}{2}\right)-2\left(-\frac{1}{2}\right)=1+1=2$