Math CentralQuandaries & Queries


Question from La, a student:

Use calculus to verify Archimedes' formula for y=9-x^2. Prove Archimedes' formula for a general parabolic arch.


I know from one of your previous questions that you are studying calculus so I am going to treat this as a calculus problem.

I start with a sketch whenever possible.

parabolic arch

y = 9 - x2

Archimedes' formula for parabolic arches says that the area under the arch is 2/3 the base times the height. This parabola intersects the x-axis ay x = ± 3 and hence the length of the base is 2 × 3 = 6 units. The height is 9 units so using, Archimedes' formula, the area under the arch is 2/3 × 6 × 9 = 36 square units.

To verify that this result is true find the are using integration.

For a general parabolic arch modify the steps above for the expression y = a - bx2.


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