210_6Q8 The classical scholar Archimedes (287-212 BC) calculated the volumes of

Question

210_6Q8

The classical scholar Archimedes (287-212 BC) calculated the volumes of many solids in a work called The Method. For many centuries this great work was lost, and all that remained were reports and excerpts in the works of other authors. From reading these reports, mathematicians were intrigued by how close Archimedes' methods were to the calculus invented by Isaac Newton and Gottfried Leibniz centuries later. Remarkably, a palimpsest of the lost work was rediscovered in the 1990s. This question concerns reworking, using modern calculus techniques, Proposition 13 of the palimpsest, which states: "The volume of an ungul is one sixth of the a volume of an enclosing cube' Consider a cube of side 2a positioned so that the centre of one face is at the origin, and take z-, y- and 2-axes parallel to the edges of the cube, as shown in the diagram below. This cube is the enclosing cube mentioned in Archimedes' result. Inscribe a circle of radius a in the top and bottom faces of the cube, and imagine cutting these out so that a cylinder of height 2a with base of radius a remains. Now imagine cutting this cylinder by the plane z 2y (shown in blue in the left-hand diagram) that passes through a diameter of the base circle and along an edge at the top of the cube. The object remaining underneath this plane resembles a horse's hoof and hence it is called an ungula; it is shown in the right-hand diagram below Use cylindrical coordinates to calculate the volume of the ungula, and hence verify Archimedes' result.

Explanation / Answer

The top of the Ungula has the equation  

=>   y+z=1

The side of the Ungula has the equation    

=>   y=x2

Volume is defined as integral of :-

   1-101-x^2 x^21-z dydzdx

=>    Volume of angula   =   V = 2/3 r2 h

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