Skills Review for Numerical Integration

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.

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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

[latex]1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20[/latex].

 

We could probably skip writing a couple of terms and write

[latex]1+2+3+4+\cdots+19+20[/latex],

which is better, but still cumbersome. With sigma notation, we write this sum as

[latex]\displaystyle\sum_{i=1}^{20} i[/latex],

 

which is much more compact.

Typically, sigma notation is presented in the form

[latex]\displaystyle\sum_{i=1}^{n} a_i[/latex]

 

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].

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Evaluate [latex]\sum _{i=2}^{5}\left(3i - 1\right)[/latex].

 

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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.

A trapezoid is shown. The top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h.

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.

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