Chapter 16, Problem 12RCC

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) Write the definition of the surface integral of a scalar function *f* over a surface *S.*

(b) How do you evaluate such an integral if *S* is a parametric surface given by a vector function **r***(u, v)*?

(c) What if *S* is given by an equation *z = g(x, y*)?

(d) If a thin sheet has the shape of a surface *S,* and the density at (*x*, *y*, *z)* is *ρ(x, y,* z), write expressions for the mass and center of mass of the sheet.

**(a)**

To determine

**To define**: The surface integral of a scalar function *f* over a surface *S*.

Explanation

Consider the surface *S* of vector function of two parameters *u* and *v*

Here,

*x*, *y*, and *z* are component functions of

Consider the parameter domain as *D* and divide the region *D* into subdivisions
*f* at that point and area at that patch. Consider a point

The surface integral of scalar function *f* over *S* is,

Here,

*m*, *n* are constants

**(b)**

To determine

**To evaluate**: The surface integral of a scalar function *f* over a surface *S*.

** (c)**

To determine

The surface integral of scalar function *f* with surface of equation

(d)

To determine

**To write**: The expressions for the mass and center of mass.

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