`f(theta) = sec^2(theta/2), [0, pi/2]` Find the average value of the function on the given interval.

Given `f(theta)=sec^2(theta/2), [0,pi/2]`

Average Value Formula`=1/(b-a)int_a^bf(x)dx`

`f_(ave)=1/(pi/2-0)int_0^(pi/2)sec^2(theta/2)(d theta)`

Integrate using the u-substitution method.

Let `u=theta/2`

`(du)/(d theta)=1/2`

`d theta=2du`

`f_(ave)=1/(pi/2)int_0^(pi/2)sec^2(u)(d theta)`

`=2/piint_0^(pi/2)sec^2(u)(2du)`

`=(2/pi)*2int_0^(pi/2)sec^2(u)du`

`=4/pi[tan(u)]_0^(pi/2)`

`=4/pi[tan((pi/2)/2)-tan(0/2)]`

`=4/pi[tan(pi/4)-tan(0)]`

`=4/pi[1-0]`

`=4/pi=1.273`

The average value is `4/pi`  or 1.273.