Area/Sigma Notation

Area/Sigma Notation Objective: To define area for plane regions with curvilinear boundaries. To use Sigma Notation to find areas.

The Area Problem • Formulas for the areas of polygons are well known. However, the problem of finding formulas for regions with curved boundaries caused difficulties for early mathematicians.

The Area Problem • Formulas for the areas of polygons are well known. However, the problem of finding formulas for regions with curved boundaries caused difficulties for early mathematicians. • The first real progress was made by Archimedes who obtained areas of regions with curved bounds by the method of exhaustion.

The Area Problem • This method, when applied to a circle, consists of inscribing a succession of regular polygons in the circle and allowing the number of sides to increase indefinitely. As the number of sides increases, the polygons tend to “exhaust” the region inside the circle, and the area of the polygons become better and better approximations of the exact area.

The Rectangle Method • We will now use Archimedes’ method of exhaustion with rectangles to find the area under a curve in the following way:

The Rectangle Method • We will now use Archimedes’ method of exhaustion with rectangles in the following way: • Divide the interval [a, b] into n equal subintervals, and over each subinterval construct a rectangle that extends from the x-axis to any point on the curve that is above the subinterval.

The Rectangle Method • For each n, the total area of the rectangles can be viewed as an approximation to the exact area under the curve over the interval [a, b]. As n increases these approximations will get better and better and will approach the exact area as a limit.

Area as a Limit; Sigma Notation • The notation we will use is called sigma notation or summation notation because it uses the uppercase Greek letter (sigma) to denote various kinds of sums. To illustrate how this notation works, consider the sum in which each term is of the form k2, where k is one of the integers from 1-5. In Sigma notation this can be written as which is read “the summation of k2 from 1 to 5”.

Sigma Notation • In general, we can look at sigma notation like this:

Example 1 • Let’s look at some examples of sigma notation.

Other Sums • Here are two other ideas we need to know • If the upper and lower limits of summation are the same, we evaluate that number once in the function. • If the function we are evaluating is a constant, we add that number to itself n times, or Cn.

Changing the limits of Summation • A sum can be written in more than one way using Sigma Notation with different limits of summation. For example, these three are all the same.

Example 2 • Express the following in sigma notation so that the lower limit of summation is 0 rather than 3.

Example 2 • Express the following in sigma notation so that the lower limit of summation is 0 rather than 3. • We need to subtract 3 from the upper and lower limits of summation. If you subtract 3 from the limits, you must add 3 to k in the function. Always do the opposite.

Changing Limits • Be careful… you add/subtract from k, not the exponent. For example, change the lower limit of summation from 3 to 1.

Properties of Sums • Theorem 6.4.1 a) b) c)

Summation Formulas • Theorem 6.4.2 a) b) c)

Example 3 • Evaluate

Definition of Area • We will now turn to the problem of giving a precise definition of what is meant by “area under a curve.” Specifically, suppose that the function f is continuous and nonnegative on the interval [a,b], and let R denote the region bounded below by the x-axis, bounded on the sides by the vertical lines x = a and x = b, and bounded above by the curve y = f(x).

Definition of Area • Divide the interval [a, b] into n equal subintervals by inserting n – 1 equally spaced points between a and b and denote those points by

Definition of Area • Each of these subintervals has width (b – a)/n, which is customarily denoted by

Definition of Area • Over each interval construct a rectangle whose height is the value of f at an arbitrarily selected point in the subinterval. Thus, if denote the points selected in the subintervals, then the rectangles will have heights and areas

Definition of Area • This can be expressed more compactly in sigma notation as: (k = # of rectangle) • We will repeat the process using more and more subdivisions, and define the area of R to be the “limit” of the areas of the approximating regions Rn as n increases without bound. That is, we define the area A as

Definition of Area • Definition 6.4.3 (Area under a curve) • If the function f is continuous on [a, b] and if f(x) > 0 for all x in [a, b], then the area under the curve y = f(x) over the interval [a, b] is defined by

Points • The values of can be chosen arbitrarily, so it is conceivable that different choices of these values might produce different values of A. Were this to happen, then the definition of area would not be acceptable. This does not happen. We will get the same area regardless of which points we choose.

Points • The three ways we will look at this is: • The left endpoint of each subinterval • The right endpoint of each subinterval • The midpoint of each subinterval

Points • To be more specific, suppose that the interval [a, b] is divided into n equal parts of length by the points and let x0 = a and xn = b. Then for k = 0, 1, 2,…,n

Points • We will look at each point as: • Left endpoint • Right endpoint • Midpoint

Example 5 • Use the definition of area using the right endpoint of each subinterval to find the area between the graph of f(x) = x2 and the interval [0, 1].

Example 5 • Use the definition of area using the right endpoint of each subinterval to find the area between the graph of f(x) = x2 and the interval [0, 1]. • We will use the same problem solving process for each point we choose. • Find

Example 5 • Use the definition of area using the right endpoint of each subinterval to find the area between the graph of f(x) = x2 and the interval [0, 1]. • We will use the same problem solving process for each point we choose. • Find • Find

Theorem 6.4.4 • Here are a few limits that you may or may not use. They can make the end of the problems easier.

Example 6 • Use the definition of area using the midpoint to find the area under the curve f(x) = 9 – x2 over the interval [0, 3].

Example 6 • Use the definition of area using the midpoint to find the area under the curve f(x) = 9 – x2 over the interval [0, 3]. • Find

Example 6 • Use the definition of area using the midpoint to find the area under the curve f(x) = 9 – x2 over the interval [0, 3]. • Find • Find

Example 6 • Let’s quickly look at this as a right endpoint and compare the two answers.

Net Signed Area • In our definition of area, we assumed that f was continuous and nonnegative over the interval [a, b]. • If f is both negative and positive over the interval, our definition no longer represents the area between the curve y = f(x) and the interval [a, b]; rather, it represents a difference of areas- the area above the x-axis minus the area below the x-axis. We call this the net signed area.

Definition 6.4.5 • Net Signed Area- If the function f is continuous on [a, b] then the net signed area A between y = f(x) and the interval [a, b] is defined by

Example 7 • Use our definition of area with the left endpoint to find the net signed area between the graph of y = x – 1 and the interval [0, 2]

Area/Sigma Notation