Question

The area bounded by the curve \( y=\sin x \) between \( \pi=\frac{\pi}{6} \) and \( \pi-\frac{5 \pi}{4} \) e the \( x- \) axis is, 

a) \( \frac{4+\sqrt{3}+\sqrt{2}}{2} \) 

b) \( \frac{4+\sqrt{3}-\sqrt{2}}{2} \) 

c) \( \frac{4-\sqrt{3}+\sqrt{2}}{2} \) 

d) None of these

Answer 1

Required bounded region

 = \(\int\limits_{π/2}^{π}sin xdx + \int\limits_\pi^{\frac{5\pi}4}(0-sin x)dx\)

 = \([-cos x]^{\pi}_{\pi/6}+[cos x]_π^{5\pi/4}\)

 = - cos π + cos π/6 + cos 5π/4 - cos π

= -(-1) + \(\frac{\sqrt3}2-\frac1{\sqrt2 }-(-1) \)

 = 2 + \(\frac{\sqrt3}2-\frac1{\sqrt2}\) square units