I'm working on this question and having trouble coming up with a solution, mainly due to the cos. I have no idea how to type sums in here so I am just going to post two screenshots of the original problem.
Basically all I have to do is solve the Reimann sum that is already provided, but it's way too complicated to work out so I have to convert it back to the integral then solve the integral. I'm having trouble finding what should be in the original integral. If I set it as cos(t) I almost get the right Riemann sum but there is still a (4pi)/(21) leftover from the delta x part of the Riemann sum, and I don't know how to change this because it's all doomed to be "trapped" inside the cos.
Alright, so that's what I originally tried, and you get sin(4pi/21) as the solution (integral on that interval of cos(x)), but that is the wrong solution. And when I actually work it out I dont' see how cos(x) can be f(x), or entirely, there must be something else. Here is my working below. The part I was talking about struggling with was that leftover 4pi/21: