(5-6)     ①   Here’s the graph:

The blue curve is the function y = cos(0.2πk):

you know that when k=0, cos (0) = 1 and when k = +/- 2.5, cos (0.5π) = 0, so it's easy enough to sketch.   

②   The red line shows the mean value of this function over the set of points spanning the range k=0 to k=0.8.

③  The area shaded in grey is the definite integral, 

④  You can see that this area will be the same as the area of the rectangle between the lines y=<y> and y=0, over the same range of k  (because that the mean y value of the points defining the top of the grey area is <y>):

The rectangle area is given by ΔxΔy  = (0.8 – 0) x (<y> - 0)  =  0.8 <y>.

 Thus, we can write:   

⑤  If we generalise the range of k, so say it’s from k1 to k2 , and we generalise the function to f(k), this becomes:

where <f(k,x)> is the mean value of the function over the range k=k1 to k=k2.

Hence, at any particular value of x, the mean value of the function f(k,x), averaged over this range of k, is given by: