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In Apostol's book:

Archimedes made the surprising discovery that the area of the parabolic segment is exactly one-third that of the rectangle; that is to say, $A=b^3/3$, where $A$ denotes the area of the parabolic segment.

Why $A=b^3/3$? Shouldn't it be $A=b^2/3$? I thought that the area of a bidimensional space was given by the square ($x^2$) not with the cube($x^3$).

Notice I'm not so sure if I can call an area of bidimensional space.

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up vote 5 down vote accepted

Well, the height is $b^2$ and the length is $b$, so $b^3 $ is the area of the rectangle. The parabola is a part of that, which is $b^3/3$. For example, if the we were dealing with $\rm cm$, we would have

$$A=b^2\;{\rm cm}\times b\;{\rm cm}=b^3 \;{\rm cm^2}$$

es expected.

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If you are thinking of graphs as physical objects with units, then you shouldn't write down the equation $y=x^2$, because the two sides have different units (distance and distance$^2$). You should write $y=x^2/k$ where $k$ has units of distance. Then the rectangle is $b \times b^2/k$ and has area $b^3/k$; the parabolic segment has area $b^3/(3k)$.

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