>If f(x)=sinx over the interval 0>x>pi/2 is wanted to be
>approximated by g(x)=px (p is a constant) what value of p
>would you choose to make:
>int.pi/2->0 f(x)dx= int.pi/2->0 g(x)dx ?

This is trivial. Just carry out the integration in

int((f - g)2)dx

to obtain a quadratic polynomial in p. Find the minimum of the parabola.

>and then what value of p would you choose to make:
>the maximum value of |f(x)-g(x)| in the interval as small as
>possible?

This falls under the Chebyshev theorem. There will be three points where the maxumum is achieved. See, e.g. P. Davis, Interpolation and Approximation, ch 7.

>and to make:
>int.pi/2->0^2dx as small as possible??