I - Area Lesson

Area

images of shapes Integral calculus is the mathematics used to define and calculate areas of regions for which no standard area formulas are known. Finding areas of regions with straight sides is relatively easy since the area formulas for triangles, A = (1/2)bh and rectangles, A = bh are well known. Even computing the area of a polygon can be accomplished by dividing it into triangles and summing the triangle areas. The difficulty arises when trying to find the area of a region with curved sides.

Trying to find the area of a circle by drawing smaller and smaller triangles never quite fills the entire circle. However, it is this approach of increasing the number of polygons and taking the limit of the sum of resulting areas that was used by the Greeks of fifth century B.C.

Sigma Notation

Sigma notation uses the Greek capital letter sigma (Σ) to denote the sum of n terms a₁, a₂, a₃,..., a_n-1, a_n and is written as equation image indicator $LaTeX: \sum_{i=1}^{n}a_{1} =a_{1}+a_{2}+...+a_{n}$ $\sum_{i=1}^{n}a_{1} =a_{1}+a_{2}+...+a_{n}$ . The variable i is the index of summation, a_i is the ith term of the sum, the lower bound of summation is 1, and upper bound of summation is n. Although any variable can be used as the index of summation, the most often used are i, j, and k. In addition, the lower bound does not have to be 1. Any integer less than or equal to the upper bound is legitimate.

Sum	Sigma Notation	Read
$a_1+a_2$	$LaTeX: \sum_{i=1}^{2}a_{1}$ $\sum_{i=1}^{2}a_{1}$	The sum of a sub i from i equals 1 to 2.
$a_2+a_3+a_4+a_5$	$LaTeX: \sum_{i=2}^{5}a_{1}$ $\sum_{i=2}^{5}a_{1}$	The sum of a sub i from i equals 2 to 5.
$a_1+a_2+...+a_n$	$LaTeX: \sum_{i=1}^{n}a_{1}$ $\sum_{i=1}^{n}a_{1}$	The sum of a sub i from i equals 1 to n.

The following properties of summation are derived using the associative, commutative, and distributive properties for real numbers.

Constant Multiple Rule	$LaTeX: \sum_{i=1}^{n}ca_{1}=c.\sum_{i=i}^{n}a_{1}$ $\sum_{i=1}^{n}ca_{1}=c.\sum_{i=i}^{n}a_{1}$ for any number c
Constant Value Rule	$LaTeX: \sum_{i=1}^{n}a_{1}=n\cdot c$ $\sum_{i=1}^{n}a_{1}=n\cdot c$ if $LaTeX: a_{1}$ $a_{1}$ has the constant value c
Sum and Difference Rule	$LaTeX: \sum_{i=1}^{n}(a_{1}\pm b_{1})=\sum_{i=1}^{n}a_{1}\pm \sum_{i=1}^{n}b_{1}$ $\sum_{i=1}^{n}(a_{1}\pm b_{1})=\sum_{i=1}^{n}a_{1}\pm \sum_{i=1}^{n}b_{1}$

Sigma notation is used to represent several formulas that are useful when finding areas:

Did you know? The famous German mathematician Carl Friedrich Gauss discovered the formula for the first n integers at age 5 equation image indicator $LaTeX: \sum_{i=1}^{n}i=\frac{n(n+1)}{2}\\ \sum_{i=1}^{n}i^2=\frac{n(n+1)(2n+1}{6}\\ \sum_{i=1}^{n}i^3=[\frac{n(n+1)}{2}]^2$ $\sum_{i=1}^{n}i=\frac{n(n+1)}{2}\\ \sum_{i=1}^{n}i^2=\frac{n(n+1)(2n+1}{6}\\ \sum_{i=1}^{n}i^3=[\frac{n(n+1)}{2}]^2$

View the Sigma Notation presentation.

Area of a Plane Region

image of Archimedes The area bounded by a curve y = f(x) on a closed interval [a, b] and one or more additional curves is described as the area of a plane region and is denoted as equation image indicator $LaTeX: A_a^bf\left(x\right)$ $A_a^bf\left(x\right)$ . Specifically, the area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles:

equation image indicator $LaTeX: A=\lim_{n \to \infty }R_n=\lim_{n \to \infty }\left[f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+...+f\left(x_n\right)\Delta x\right]$ $A=\lim_{n \to \infty }R_n=\lim_{n \to \infty }\left[f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+...+f\left(x_n\right)\Delta x\right]$

This area written using sigma notation is equation image indicator $LaTeX: A= \sum_{i=1}^{n}f\left(x_1\right)\Delta x=\left[f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+...+f\left(x_n\right)\Delta x\right]$ $A= \sum_{i=1}^{n}f\left(x_1\right)\Delta x=\left[f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+...+f\left(x_n\right)\Delta x\right]$ .

(Archimedes Caption: Did You Know? The ancient Greeks and Archimedes (287-212 B.C.), considered the greatest applied mathematician of antiquity, used the exhaustion method to determine formulas for the areas of regions bounded by conics. This method is a limiting process in which area is squeezed between two polygons; one inscribed in the region and one circumscribed about the region.

View the presentation on using the limit process to find the exact area under a curve using limits.

Lower Sums and Upper Sums

The area of a plane region can be computed by summing the areas of rectangles subdividing the interval [a, b] into n subintervals, each of width equation image indicator $LaTeX: \Delta x=\frac{\left(b-a\right)}{n}$ $\Delta x=\frac{\left(b-a\right)}{n}$ . The lower sum is the area of inscribed rectangles denoted by $LaTeX: s\left(n\right)= \sum_{i=1}^{n}f\left(m_i\right)\Delta x$ $s\left(n\right)= \sum_{i=1}^{n}f\left(m_i\right)\Delta x$ and produces an underestimate of the area. The upper sum is the area of circumscribed rectangles denoted by equation image indicator $LaTeX: S\left(n\right)= \sum_{i=1}^{n}f\left(M_i\right)\Delta x$ $S\left(n\right)= \sum_{i=1}^{n}f\left(M_i\right)\Delta x$ and produces an overestimate of the area. Finding an explicit value for the area under a curve by summing areas of rectangles is called the rectangle approximation method (RAM).

Left, Right, and Midpoint Approximations

Creating rectangles for use in the rectangle approximation method involves choosing the value of the function f at the left endpoint, f(x_i-1), the value of f at the right endpoint, f(x_i),or the value of f at the midpoint, f((x_i-1+x_i)/2), as the height of each rectangle. The base of the rectangle is the width of each subinterval, equation image indicator $LaTeX: \Delta x=\frac{\left(b-a\right)}{n}$ $\Delta x=\frac{\left(b-a\right)}{n}$ . Notation for the RAM results using n subintervals and left endpoints is LRAM_n f, right endpoints is RRAM_n f, and midpoints is MRAM_n f. Click HERE to view the presentation on using rectangle approximation methods to approximate the area under a specific curve. Links to an external site.

Links to an external site.Explore more information relating acceleration, velocity, and position to area under their graphs by CLICKING HERE. Links to an external site.

Area Practice

Area: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of areas and distance.

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