The Quadrature of the Parabola
Archimedes used geometric series for computing the areas that are enclosed by a straight line and parabola. The method was to dissect an area into an infinite number of triangles.
The Theorem states that the complete area in the parabola is 4/3 of the area of a blue triangle. He found that every green triangle has 1/8 of an area of the blue triangle and the yellow triangle has 1/8 if the area of a green triangle. Assuming that the blue triangle has a total area of 1 then the total area is the infinite series:
The first term says the area of a blue triangle, in the second term there are areas of the green triangles and in the third term there are areas of the four yellow triangles. When you simplify the fraction, you shall get,
This is a geometric series with a common ratio ¼ and the fractional part is 1/3. This is stated in our help with assignment online on Geometric Series topics.
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