Learning Objectives

Use sigma (summation) notation to calculation sums and powers that integers.Use the amount of rectangular areas to approximate the area under a curve.Use Riemann sums to approximate area.

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Archimedes to be fascinated v calculating the areas of miscellaneous shapes—in various other words, the lot of room enclosed by the shape. He offered a process that has concerned be well-known as the method that exhaustion, which provided smaller and also smaller shapes, the areas of which could be calculate exactly, to to fill an irregular region and thereby attain closer and closer approximations to the full area. In this process, an area bounded by curves is filled through rectangles, triangles, and shapes with specific area formulas. These areas are climate summed to almost right the area the the curved region.

In this section, we develop techniques to approximate the area in between a curve, identified by a role

*
and also the
*
-axis on a closeup of the door interval
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favor Archimedes, we an initial approximate the area under the curve using forms of known area (namely, rectangles). By utilizing smaller and also smaller rectangles, we gain closer and closer approximations come the area. Taking a limit permits us to calculation the precise area under the curve.

Let’s begin by introducing some notation to make the calculations easier. We then take into consideration the case when

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is constant and nonnegative. Later on in the chapter, we relax several of these restrictions and also develop approaches that use in more general cases.


Sigma (Summation) Notation

As mentioned, we will certainly use forms of known area to almost right the area of one irregular region bounded by curves. This procedure often requires including up long strings that numbers. To make it less complicated to write down these an extensive sums, us look in ~ some brand-new notation here, called sigma notation (also recognized as summation notation). The Greek capital letter

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sigma, is provided to express lengthy sums of values in a compact form. For example, if we desire to include all the integers native 1 to 20 without sigma notation, we need to write


*

We might probably skip creating a pair of terms and also write


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which is better, yet still cumbersome. Through sigma notation, we compose this amount as


*

which is much much more compact.

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Typically, sigma notation is gift in the form


*

where

*
explains the state to be added, and also the i is dubbed the index. Each term is evaluated, then we sum all the values, beginning with the value once
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and also ending through the value as soon as
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for example, one expression choose
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is interpreted as
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keep in mind that the index is offered only to store track the the state to it is in added; the does not aspect into the calculation of the amount itself. The table of contents is because of this called a dummy variable. We have the right to use any type of letter we favor for the index. Typically, mathematicians usage i,
*
,
*
,
*
, and
*
for indices.