Some ofthe geometric formulas we take for granted today were first derived by methods that anticipate some ofthe methods of calculus. The Greek mathematician Archimedes (ca. 287 212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area ofthe circle as the number of sides ofthe polygon increased. He never came up with the idea of a limit, but we can use this idea to see what his geometric constructions could have predicted about the limit. We can estimate the area of a circle by computing the area of an inscribed regular polygon. Think ofthe regular polygon as being made up of n triangles. By taking the limit as the centex angle of these mangles goes to zero, you can obtain the area ofthe circle. To see this, carry out the following steps: 2. Find an expression for the area of the n-sided polygon in terms of r and θ
Some ofthe geometric formulas we take for granted today were
first derived by methods that anticipate some ofthe methods of
calculus. The Greek mathematician Archimedes (ca. 287 212;
BCE) was particularly inventive, using polygons inscribed within
circles to approximate the area ofthe circle as the number of
sides ofthe
a limit, but we can use this idea to see what his geometric
constructions could have predicted about the limit.
We can estimate the area of a circle by computing the area of an
inscribed regular polygon. Think ofthe regular polygon as being
made up of n triangles. By taking the limit as the centex angle of
these mangles goes to zero, you can obtain the area ofthe circle.
To see this, carry out the following steps:
2. Find an expression for the area of the n-sided polygon in terms of r and θ
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