Archimedes / Quadrature by parts

Before a birth of calculus, how to solve areas/volumes of a solid figure made up of smooth curves was referred to as quadrature by parts. Different from calculus, this method depends on figure types so as to successfully achieve clever solutions specific to the types.

In the times of the ancient Greek, Archimedes (278-212 B.C.) demonstrated "An area of a parabola is 4/3 times as large as an area of .

The demonstration is as follows.
Draw the two tangents to the parabola at points and , and the point of intersection between the two is defined as .
There is a third tangent parallel to side , and the point of intersection between this tangent and the parabola is defined as .
Produce line to , and the point of intersection between this and the side is defined as .
Then, point represents the middle point of side .
Therefore an area of

Next, Draw a tangent to the parabola at point , and the point of intersection between the two is defined as .
There is a third tangent parallel to side , and the point of intersection between this tangent and the parabola is defined as .
Produce line to , and the point of intersection between this and the side is defined as .

Suppose an area of is , then an area of .

Taking into account another triangle in the side of point yields .
Such procedure is to be endlessly repeated. However, only infinite times can cover all over the area .
Archimedes thought as follows:




Since is arbitrary, cannot be but .

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