Skills Review for Approximating Areas

Learning Outcomes

  • Calculate the area of a rectangle
  • Use summation notation

In the Approximating Areas section, we will estimate the area under a curve by dividing the region under the curve into rectangles. Here we will review how to find the area of a rectangle and how to expand summation notation.

Find the Area of a Rectangle

The area of a rectangle can be found using the following formula:

[latex]A=lw[/latex]

where [latex]l[/latex] is the length and [latex]w[/latex] is the width of the rectangle.

Note: Sometimes the area of a rectangle can be found using the formula [latex]A=bh[/latex] where [latex]b[/latex] is the base and [latex]h[/latex] is the height of the rectangle.

Example: Finding The Area of a Rectangle

Find the area of a rectangle with a length of 3 inches and a width of 2 inches.

Example: Finding The Area of a Rectangle

Find the area of a rectangle with a base of 2 centimeters and a height of 7 centimeters.

Try It

Find the area of a rectangle with a base of 7 centimeters and a height of 9 inches.

Expand Sigma (Summation) Notation

Summation notation is used to represent long sums of values in a compact form. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms of the sum. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to the lower limit of summation, which is the number used to generate the first term of the sum. The number above the sigma, called the upper limit of summation, is the number used to generate the last term of the sum.

If we interpret the given notation, we see that it asks us to find the sum of the terms in the series [latex]{a}_{i}=2i[/latex] for [latex]i=1[/latex] through [latex]i=5[/latex]. We can begin by substituting the terms for [latex]i[/latex] and listing out the terms.

[latex]\begin{array}{l} {a}_{1}=2\left(1\right)=2 \\ {a}_{2}=2\left(2\right)=4\hfill \\ {a}_{3}=2\left(3\right)=6\hfill \\ {a}_{4}=2\left(4\right)=8\hfill \\ {a}_{5}=2\left(5\right)=10\hfill \end{array}[/latex]

We can find the sum by adding the terms:

[latex]\displaystyle\sum _{i=1}^{5}2i=2+4+6+8+10=30[/latex]

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]\displaystyle\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]\displaystyle\sum _{i=3}^{7}{i}^{2}[/latex].

Try It

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

 

Try It