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Archimedes' area formula for parabolic arches Archimedes $$( 287 - 212$$ B.c.), inventor, military engineer, physicist, and the greatest mathematician of classical times in the Western world, dis- covered that the area under a parabolic arch is two-thirds the base times the height. Sketch the parabolic arch $$y = h - ( 4 h / b ^ { 2 } ) x ^ { 2 }$$ , $$- b / 2 \leq x \leq b / 2$$ , assuming that $$h$$ and $$b$$ are positive. Then use calculus to find the area of the region enclosed between the arch and the $$x$$ -axis.
$$\frac { 2 } { 3 } b h$$