Approximating Finite Sums with Integrals

In many applications of calculus, integrals are used to approximate finite sums—the reverse of the usual procedure of using finite sums to approximate integrals.

For example, let’s estimate the sum of the square roots of the first n positive integers, The integral

is the limit of the upper sums

Therefore, when n is large, will be close to 2/3 and we will have

The following table shows how good the approximation can be.

Let

To calculate show that and interpret as an approximating sum of the integral

(Hint: Partition [0, 1] into n intervals of equal length and write out the approximating sum for inscribed rectangles.)