Learning Outcomes
- Complete a table values with solutions to an equation
- Use sigma (summation) notation to calculate sums and powers of integers
- Find the area of a trapezoid
In the Numerical Integration section, we will estimate the value of definite integrals using a variety of estimation rules. Here we will review some essential topics that will help us to better understand the various estimation rules.
Complete a Table of Function Values
A table of values can be used to organize the y-values or function values that result from plugging specific x-values into a function’s equation.
Suppose we want to determine the various values of the equation [latex]f(x)=2x - 1[/latex] at certain values of x. We can begin by substituting a value for x into the equation and determining the resulting value of the function. The table below lists some values of x from –3 to 3 and the resulting function values.
| [latex]x[/latex] | [latex]f(x)=2x - 1[/latex] |
| [latex]-3[/latex] | [latex]f(-3)=2\left(-3\right)-1=-7[/latex] |
| [latex]-2[/latex] | [latex]f(-2)=2\left(-2\right)-1=-5[/latex] |
| [latex]-1[/latex] | [latex]f(-1)=2\left(-1\right)-1=-3[/latex] |
| [latex]0[/latex] | [latex]f(0)=2\left(0\right)-1=-1[/latex] |
| [latex]2[/latex] | [latex]f(2)=2\left(2\right)-1=3[/latex] |
| [latex]3[/latex] | [latex]f(3)=2\left(3\right)-1=5[/latex] |
We can look for trends among our function values by looking at the table. For example, in this case, as x-values increase, so do the y-values. Also, it seems reasonable to assume, based on the table, an x-value of 1 would result in a function value of 1. You can verify this for yourself by plugging 1 into [latex]f(x)=2x - 1[/latex].
Example: Completing a table of function values
Create a table of function values for [latex]f(x)=-x+2[/latex]. Use various integers from -5 to 5 as the x-values you plug into the function.
Try It
Try It
Use Sigma (Summation) Notation
Summation notation (also known as summation notation) is used to make it easier to write lengthy sums. The Greek capital letter [latex]\Sigma[/latex], sigma, is used to express long sums of values in a compact form. For example, if we want to add all the integers from 1 to 20 without sigma notation, we have to write
We could probably skip writing a couple of terms and write
which is better, but still cumbersome. With sigma notation, we write this sum as
which is much more compact.
Typically, sigma notation is presented in the form
where [latex]a_i[/latex] describes the terms to be added, and the [latex]i[/latex] is called the index. Each term is evaluated, then we sum all the values, beginning with the value when [latex]i=1[/latex] and ending with the value when [latex]i=n[/latex]. For example, an expression like [latex]\displaystyle\sum_{i=2}^{7} s_i[/latex] is interpreted as [latex]s_2+s_3+s_4+s_5+s_6+s_7[/latex]. Note that the index is used only to keep track of the terms to be added; it does not factor into the calculation of the sum itself. The index is therefore called a dummy variable. We can use any letter we like for the index. Typically, mathematicians use [latex]i[/latex], [latex]j[/latex], [latex]k[/latex], [latex]m[/latex], and [latex]n[/latex] for indices.
A General Note: Summation Notation
The sum of the first [latex]n[/latex] terms of a series can be expressed in summation notation as follows:
[latex]\sum _{i=1}^{n}{a}_{i}[/latex]
This notation tells us to find the sum of [latex]{a}_{i}[/latex] from [latex]i=1[/latex] to [latex]i=n[/latex].
[latex]k[/latex] is called the index of summation, 1 is the lower limit of summation, and [latex]n[/latex] is the upper limit of summation.
Example: EXpanding Summation Notation
Evaluate [latex]\sum _{i=3}^{7}{i}^{2}[/latex].
Try It
Evaluate [latex]\sum _{i=2}^{5}\left(3i - 1\right)[/latex].
Try It
Find the Area of a Trapezoid
A trapezoid is a four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base [latex]b[/latex] and the larger base [latex]B[/latex]. The height, [latex]h[/latex], of a trapezoid is the distance between the two bases as shown in the image below.
The formula for the area of a trapezoid is:
[latex]Area_{trapezoid}=\dfrac{1}{2}h(b+B)[/latex]
Example: Finding the Area of a Trapezoid
Find the area of a trapezoid whose height is 6 inches and whose bases are 14 and 11 inches.
