   Chapter 16, Problem 3RCC

Chapter
Section
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, atb , with parametric equations,

x=x(t)y=y(t)

Consider a vector equation as r(t)=x(t)i+y(t)j and the first derivative (r(t)) of vector function is continuous, and not equal to zero, r(t)0 . Divide the parametric interval of curve C which is [a,b] , into the equal width of n subintervals as [ti1,ti] and subarcs of curve C as Δs1,Δs2,...,Δsn

(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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