Question

How do you use Riemann sums to evaluate the area under the curve of `f(x)=cosx+0.5` on the closed interval [0,2pi], with n=pi rectangles using midpoints?

Answer 1

`pi * f((0+pi)/2)+pi * f((pi+2pi)/2)=pi`, so the area under the curve is about `pi`. (In fact, it is exactly `pi`).

Explanation:

Since `n=pi`, we will be measuring rectangles with length `pi`. Since we're operating on the interval `[0,2pi]`, we will only have two rectangles.

The first rectangle is on the interval `[0,pi]`. The midpoint of `x=0` and `x=pi` is `x=(0+pi)/2=pi/2`.

At `x=pi/2` we have a rectangle with a base of `pi` and height of `f(pi/2)=cos(pi/2)+1/2=1/2`.

This rectangle has an area of `A=pi(1/2)=pi/2`.

The next rectangle will be on the interval `[pi,2pi]`, of which the midpoint is located at `x=(3pi)/2`.

The height of this rectangle is `f((3pi)/2)=cos((3pi)/2)+1/2=1/2`.

The area of the rectangle with base `pi` and height `1/2` is again `A=pi(1/2)=pi/2`.

The two rectangles we've found both have area `pi/2`, so the total area of both rectangles is `pi`, the approximate area under the curve.