Problem

Write the expression

as a definite integral.

Solution

Inside the limit, we have a sum of areas of rectangles, or their negatives if the function values are negative. These rectangles approximate a function. Our job is to find out which function they approximate, as well as the starting and ending points of this approximation on the x axis.

The area of rectangle i is

Let's assume 2p/n is the width of each rectangle (or its corresponding subinterval on the x axis) and

is the height. If we have n rectangles, each of width 2p/n, then the total width of the interval equals 2p.

When i = 0, we're evaluating cos(p), so p is our starting point on the x axis, and p + 2p = 3p is our ending point. Because each subinterval has width 2p/n, for the ith subinterval we find that

is cos(x) evaluated at the leftmost point of that subinterval. So f(x) = cos(x), and our bounds are p and 3p.

All together,

This can be generalized to apply to any limit of sums in this form: