c) Explain why Guldin\'s Principle works by setting up the integrals required to

Question

c) Explain why Guldin's Principle works by setting up the integrals required to find the volume of the

solid of revolution and comparing these to Guldin's recipe.

d) Illustrate the use of GP by determining the volume of a torus with radii R(big) and r(small).

e) Use the formula for the volume of a sphere with radius r and GP to find the center of gravity for a semi-disc.

(For convenience sake, assume that the bounding diameter lies on the x-axis. Verify your answer using calculus.)

f) Use part e. and GP to find the volume of the outside part of the torus

(i.e. of the solid formed by those points inside the torus that are more than Rbig units away from the axis of the torus).

Explanation / Answer

c) after an interesting preface concerning regular polygons, and containing remarks upon the hexagonal form of the cells of honeycombs, Pappus addresses himself to the comparison of the areas of different plane figures which have all the same perimeter (following Zenodorus's treatise on this subject), and of the volumes of different solid figures which have all the same superficial area, and, lastly, a comparison of the five regular solids of Plato. Incidentally Pappus describes the thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by Archimedes, and finds, by a method recalling that of Archimedes, the surface and volume of a sphere.

d)

Given that:

Outer radius(R)
Inner radius(r)

Substitute the given values in the formula:

Volume of torus = 2 × π2 × R × r2

e)

)

e z+d

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