Calculate work in clear way

Question

Find the average value of the function 𝑓(𝑥)=−2sin𝑥f(x)=−2sin⁡x on the given intervals.

a) Average value on [0,𝜋/2][0,π/2]:   

b) Average value on [0,𝜋][0,π]:   

c) Average value on [0,2𝜋][0,2π]:   

Answer 1

a) -4/(pi)

b) -4/(pi)

c) zero.

The reasons are that each average value is defined as the definite integral of the function -2*sin(x) on the interval in question divided by the distance between the endpoints of that interval. The indefinite integral of -2*sin(x) is 2*cos(x)+C because the derivative of cos(x) is -sin(x). So, for a): 2*cos ((pi)/2) - 2*cos (0) = 0 - 2 = -2, and that divided by (pi)/2 is -4/(pi). For b): 2*cos (pi) - 2*cos (0) = (-2) - 2 = -4 divided by pi. And for c): 2*cos (2*pi) - 2*cos (0) = 2 - 2 = 0.