Chapter 15.5, Problem 22E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

If you attempt to use Formula 2 to find the area of the top half of the sphere *x*^{2} + *y*^{2} + *z*^{2} *= a*^{2}, you have a slight problem because the double integral is improper. In fact, the integrand has an infinite discontinuity at every point of the boundary circle *x*^{2} + *y*^{2} = *a*^{2}*.* However, the integral can be computed as the limit of the integral over the disk *x*^{2} + *y*^{2} ≤ *t*^{2} as t → *a*^{-}. Use this method to show that the area of a sphere of radius *a* is 4π*a*^{2}*.*

To determine

**To show:** The area of the sphere of radius *a* is equal to

Explanation

**Formula used:**

The surface area with equation

Here, *D* is the given region.

If
*R* given by

If

**Given:**

The sphere,

**Calculation:**

Solve the given equation as shown below.

The partial derivatives

Then, by the formula mentioned above, the area of the surface is given by,

Use polar coordinates to solve

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