Mathematics

Problem

For

0\le x\le \pi

let

u(x)=\cos x

and

v(x)=1-\dfrac{x}{\pi}

. (i) Find the average value of

u

[0,\pi]

. (ii) Find the average value of

v

[0,\pi]

. (iii) Compare the two average values obtained and state which is greater. (iv) By how much does the greater average exceed the smaller one?