Approximating Finite Sums with IntegralsIn
Chapter 5, Problem 27AAE(choose chapter or problem)
Problem 27AAE
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.
a. Show that the area of an n-sided regular polygon in a circle of radius r is
b. Find the limit of Is this answer consistent with what you know about the area of a circle?
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