Examples

Surface area of a sphere

Using the fact that a sphere is given by revolving a semicircle, verify the formula for the surface area of a sphere.

Solution

(1) Describe sphere as surface of revolution:

Upper semicircle:

As function of :

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(2) Surface area formula:

Our bounds are and . Function is :

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(3) Work out integrand:

Power rule and chain rule:

Therefore:

Integrand:

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(4) Compute integral:

This is the expected surface area formula: .