Chapter 16, Problem 3RCC

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) Write the definition of the line integral of a scalar function *f* along a smooth curve *C* with respect to arc length.

(b) How do you evaluate such a line integral?

(c) Write expressions for the mass and center of mass of a thin wire shaped like a curve *C* if the wire has linear density function *ρ*(*x*, *y*).

(d) Write the definitions of the line integrals along *C* of a scalar function *f* with respect to *x*, *y*, and z.

(e) How do you evaluate these line integrals?

**(a)**

To determine

**To define**: The line integral of scalar function *f* along smooth curve *C*.

Explanation

The integrals which are done over a curve instead of a interval are referred as line integrals.

Consider a smooth curve *C*,

Consider a vector equation as
*C* which is
*n* subintervals as

**(b)**

To determine

**To evaluate**: The line integral of scalar function *f*.

**(c)**

To determine

**To write**: The expressions for mass and center of mass of a thin wire shaped like cure *C*.

**(d)**

To determine

**To define**: The line integrals of scalar function *f* along *C* with respect to *x*, *y*, and *z*.

**(e)**

To determine

**To evaluate**: The line integrals of scalar function *f* along *C* with respect to *x*, *y*, and *z*.

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