What is the area of the region bound by the graphs of f(x) and g(x) between those points of intersection? (Multiple Choice)

Question

The graphs of the functions f(x)=sin(x) and g(x)=1/2 intersect at 2 points on the interval 0<x<pi

What is the area of the region bound by the graphs of f(x) and g(x) between those points of intersection?

A. sqrt3 - pi/3

B. 2 - pi/2

C. pi/3

D. pi/2

Answer 1

The sine function is 1/2 at x = pi/5 and x = 5 * pi/6 ( or 30 and 150 degrees)

integral ( sin x - 1/2) = -cos x - (1/2) x

x=5*pi/6 ---> - cos(5 * pi/6) - (1/2)(5 * pi/6)

= - ( -sqrt(3)/2) - 5*pi/12

= sqrt(3)/2 - 5 * pi/12

= (6 * sqrt(3) - 5 *pi) / 12

x = pi/6 ---> -cos(pi/6) - (1/2)(pi/6)

= -sqrt(3)/2 - pi/12

= (-6*sqrt(3) - pi) / 12

[(6 * sqrt(3) - 5 * pi ) - ( -6 * sqrt(3) + pi ]/ 12 =

(12 * sqrt(3) - 4 * pi ) / 12

(3 * sqrt(3) - pi) / 3