Chapter 16.6, Problem 50E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Find the area of the surface.

**50.** The part of the sphere *x*^{2} +* y*^{2} + *z*^{2} = *b*^{2} that lies inside the cylinder *x*^{2} + *y*^{2} = *a*^{2}, where 0 *< a < b*

To determine

**To find:** The area of the part of the sphere

Explanation

**Given data:**

The equation of the part of the sphere is given as follows.

The required sphere lies inside the cylinder

Here, the limit of radius of the cylinder

**Formula used:**

Write the expression to find the surface area of the plane.

As the cylinder encloses the sphere separate portions of the sphere in the upper and lower halves, the required area is the two times of the area determined with the formula in equation (1).

Write the expression to find required original area of the sphere as follows.

Write the equation of part of the sphere as follows.

Rewrite the expression as follows.

**Calculation of**
**:**

Take partial derivative for equation (3) with respect to *x*.

**Calculation of**
**:**

Take partial derivative for equation (3) with respect to *y*.

**Calculation of surface area of plane:**

Substitute

Simplify the expression as follows.

Consider the parametric equations for the cylinder

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