Chapter 16, Problem 12RCC

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

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 (r(u,v)) is expressed as,

r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k (1)

Here,

x, y, and z are component functions of r .

Consider the parameter domain as D and divide the region D into subdivisions Sij . Surface integral of scalar function is a Riemann sum of product of value of f at that point and area at that patch. Consider a point Pij and corresponding area of that patch as ΔSij .

The surface integral of scalar function f over S is,

Sf(x,y,z)dS=i=1mn=1nf(Pij)ΔSij

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 z=g(x,y) .

(d)

To determine

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

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