The region bounded by the x-axis?

Question

The region bounded by the x-axis and the part of the graph of y=cos x between x=-pi/2 and x=pi/2 is separated into two regions by the line x=k. If the area of the region for -pi/2≤x≤k is three times the area of the region for k≤x≤pi/2, what's k?

 

Answer: pi/6

 

 

Answer 1

∫_(-pi/2)^(pi/2) cos(x)dx = 2

 

∫_(-pi/2)^k cos(x)dx + ∫_k^(pi/2) = 2

 

∫_(-pi/2)^k cos(x)dx =  (1/2) = 3*∫_k^(pi/2)cos(x)dx = 3*(1/2)

 

sin(k) - sin(-pi/2) = 3sin(pi/2) - 3sin(k)

 

sin(k) = -3[sin(k)]

 

sin(k) = -3[sin(k)]

 

sin(k) + 3[sin(k)] = 2

 

4[sin(k)] = 2

 

sin(k) = (1/2)

 

k = (pi/6) radians or 30º