Chapter 15.5, Problem 12E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Find the area of the surface.

**12**. The part of the sphere *x*^{2} + *y*^{2} + *z*^{2} = 4z that lies inside the paraboloid z = *x*^{2}+ *y*^{2}

To determine

**To find:** The area of given surface.

Explanation

**Given:**

The function is the part of the sphere,

The region *D* lies inside the paraboloid

**Formula used:**

The surface area with equation

Here, *D* is the given region.

If *R* given by

**Calculation:**

Solve the given equations.

This yields,

Simplify further as follows.

Convert the given problems into the polar coordinates to make the problem easier. By the given the conditions, it is observed that, *r* varies from 0 to

Obtain the partial derivatives of *f* with respect to *x* and *y*

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