Let . The average value of on an interval of length is given to be . The left…

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

Let

h(x)=\sin x

. The average value of

h

on an interval of length

\pi

is given to be

\dfrac{1}{\pi}

. The left endpoint of the interval is

a

and

0\le a\le 2\pi

. (i) Write the definite integral representing the average value of

h

[a, a+\pi]

. (ii) Simplify the integral in part (i) so that the result involves

\cos a

only. (iii) Hence determine all possible values of

a

in the range

0\le a\le 2\pi

that satisfy the given average value condition.