To make it easier to write down these lengthy sums, we look at some new notation here, called sigma notation (also known as summation notation). This process often requires adding up long strings of numbers. Sigma (Summation) NotationĪs mentioned, we will use shapes of known area to approximate the area of an irregular region bounded by curves. Later in the chapter, we relax some of these restrictions and develop techniques that apply in more general cases. We then consider the case when f ( x ) f ( x ) is continuous and nonnegative. Let’s start by introducing some notation to make the calculations easier. Taking a limit allows us to calculate the exact area under the curve. By using smaller and smaller rectangles, we get closer and closer approximations to the area. Like Archimedes, we first approximate the area under the curve using shapes of known area (namely, rectangles).
In this section, we develop techniques to approximate the area between a curve, defined by a function f ( x ), f ( x ), and the x-axis on a closed interval. These areas are then summed to approximate the area of the curved region. In this process, an area bounded by curves is filled with rectangles, triangles, and shapes with exact area formulas.
He used a process that has come to be known as the method of exhaustion, which used smaller and smaller shapes, the areas of which could be calculated exactly, to fill an irregular region and thereby obtain closer and closer approximations to the total area.
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