Question

Evaluate the integral
${\int }_{\pi /6}^{5\pi /6}\sqrt{4-4{\sin }^{2}t}dt$

• A. $0$
• B. $2$
• C. $1$
• D. $4$

Answer 1

The correct option is B $2$

$2{\int }_{\frac{\pi }{6}}^{\frac{5\pi }{6}}|cost|dt$
$2[2{\int }_{\frac{\pi }{6}}^{\frac{\pi }{2}}costdt-2{\int }_{\frac{\pi }{2}}^{\frac{5\pi }{6}}costdt]$
$2[\left(sint2{\int }_{\frac{\pi }{6}}^{\frac{\pi }{2}}\right)-(sint\left)2{\int }_{\frac{\pi }{2}}^{\frac{5\pi }{6}}\right]$
$=2[(1-\frac{1}{2})-(\frac{+1}{2}-1)]$
$=2\left[1\right]=2$