Learning ObjectivesUse 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
Let’s begin by introducing some notation to make the calculations easier. We then take into consideration the case when
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
We might probably skip creating a pair of terms and also write
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