   Chapter 8, Problem 1RCC

Chapter
Section
Textbook Problem

(a) How is the length of a curve defined? (b) Write an expression for the length of a smooth curve given by y = f(x), a ≤ x ≤ b. (c) What if x is given as a function of y?

(a)

To determine

To Define: The length of the curve.

Explanation

Draw the Figure 1 using the procedure as follows:

• Draw a curve C for equation y=f(x) .
• Here,f is a continuous function and axb .
• Divide the curve as a polygonal curve with n number of subintervals at the interval [a,b] .
• The endpoint is a=x0<x1<x2<...<xn=b . The equal width of subinterval as Δx .
• The point Pi(xi,yi) lies on the curve for yi=f(xi) and the vertices of polygon P0,P1,.....Pn

Show the curve as in Figure 1. Refer Figure 1,

If the limit exists, the length of the curve C with equation y=f(x) , for axb as the limit of the lengths of these inscribed polygons.

L=limni=1n|Pi1Pi|

Thus, the length of the curve is defined.

(b)

To determine

To Express: The length of a smooth curve given by y=f(x),axb .

The expression for the length of a smooth curve is L=ab1+[f(x)]2dx

Explanation

Consider the curve is a straight line between the coordinates (x1,y1) and (x2,y2) . Then the length of curve can be expressed by using Pythagorean Theorem.

L=(Δx)2+(Δy)2 (1)

Here, Δx=x2x1 and Δy=y2y1

Suppose the Equation of curve formed by y=f(x) or x=f(x) where axb, then the Equation(1) can be expressed as

L=ab(dx)2+[f(x)dx]2L=ab1+[f(x)]2dx

Thus, the length of a smooth curve given by y=f(x),axb is defined.

(c)

To determine

To Explain: The equation of curve has x as a function of y.

If x as a function of y the expression of length of curve is L=cd1+[g(y)]2dy

Explanation

Consider x=g(y) where cyd , then f(x)=g(y) therefore the length of curve can be expressed as,

L=cd1+[g(y)]2dy

Thus, the equation of curve has x as a function of y is explained.

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