How to Evaluate and Solve Riemann Sums for Integration: Step-by-Step Guide

[Samuelb88]

Homework Statement

Evaluate and give an exact answer.

$$\int_{0}^{\frac{Pi}{2}\right)}cos^2(2*Pi*n*x)dx$$ where n is a positive integer.

Homework Equations

The Attempt at a Solution

$$(change)x=\frac{b-a}{n}\right)=\frac{Pi}{2n}\right)$$
and...
$$x_i=a+i(change)x$$

I'm not sure how to write a Riemann Sum in latex code, so I'll be using "S" as the notation for the Riemann Sum (i=1, n): f(x_i)*(b-a)/n

I used the limit as n -> oo definition of an integral:

$$\int_{0}^{\frac{Pi}{2}\right)}cos^2(2*Pi*n*x)dx=lim(n->oo)S[f(x_i)(\frac{b-a}{n}\right))]$$

Next I substituted the values of $$x_i$$:

$$lim(n->oo)S[f(\frac{ib}{n}\right))(\frac{Pi}{2n}\right))]$$

Then evaluated $$f(x_i)$$:

$$lim(n->oo)S[cos^2(2*Pi*n*\frac{Pi}{2n}\right)]$$

Here is where i get a bit confused. In the example with n-subintervals that I worked through in my book, it substituted the value of the Riemann sum of i using the power of sums formula, however, I end up with "i" inside of the cosine function, and am not sure how to finish evaluating the integral.

Other examples in the book had the value of n defined, thus making the integration process much more simple.

Am I even doing the right thing? Bare with me, my integration is self taught.

 

[HallsofIvy]

Why are you trying to change to Riemann sums? That is, at best, a very difficult way of evaluating integrals! (Though excellent for thinking how to set up integrals for applications.)

Do you know the anti-derivatives of sine and cosine themselves? If so you can use the trig identity $cos^2(\theta)= (1/2)(1+ cos(2\theta)$.