**Sub-Interval Summation**

The integral is used to find the area of a function. It is an additive process. The two basic principles for this process are (1) subdividing the function by increasingly small intervals and (2) using the definite integral to offer a closer approximation of the function area. For (1), the Riemann sum equation can be used to denote a quotient of range to n, the number of arbitrary subintervals:

In this broad definition, ∆x summation can be thought of as descendent from method of exhaustion. Described by Archimedes, it is a concept of measurement by filling a shape or space (Apostol, 1967). The following sections describe integrals to 'fill' a space under a curve and are based on Jordon measure.

**The Riemann Sum Equation**

The sum of subintervals exists in a two dimension Cartesian plane. For a continuous and nonnegative function f for the interval [a, b] divided into subintervals, the space becomes demarcated by a series of rectangles each with amplitude at f. The representative points on the function curve x1, x2, …,xn, equal the subinterval height. The area under f is defined by a definite integral:

Where

How the point of representation intersects with f and the sub-interval area effects how the Riemann sum will approach the function area. For cases where the sub-intervals intersect with the (nonnegative) function at the right- end point of the rectangle. This point can be arbirarily changed to the mid-point or left- end point of the rectangle. The effect of this control can be thought of as the Riemann sum approximation approaching the function area from above or below the function line.

**The Area between Two Functions**

The fundamental theorem of calculus states that for function f, the definite integral with limit a to b, the anti-derivative F is scalable from a to b:

#### Exercise 1

Given:

##### Solution

Solution:

The theorem uses the definite integral to find the area of function f(x) between a and b. This technique can be expropriated to find the area between two functions.

Area Between Functions 'f' and 'g' |
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Given two functions f and g (Figure 1), where f(x) is greater than g(x), an area exists between the two. The following equation defines the area between points a and b:

Figure 1 shows an instance of mirroring the integral to the function value in relation to the other at the point of intersection.

**The Area of a Radial Set**

This section has been adapted from Apostol, Tom M. "Calculus" One-Variable Calculus, with an Introduction to Linear Algebra, vol. 1. Blaisdell Publishing. 1967. (pp.109-110):

Area Defined by a Radial Set |
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The area between a and b where 0 ≤ b – a ≤ 2π, for non-negative function f, has an area defined by a radial set S of a step function s. The circular arc is the summation of n subintervals (