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) Sphere as surface of revolution.

Sphere of radius given by revolving upper semicircle.

Upper semicircle:

Upper semicircle as function of :

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(2) Surface area formula.

Bounds are and .

Function is

Plug data into formula:

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(3) Compute .

Power rule and chain rule:

Algebra:

Squaring:

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(4) Compute integrand.

Compute :

Integrand factors become:

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(5) Compute integral.

Surface area again:

This is the desired surface area formula .