﻿ ﻿LENGTH OF A CURVE - Integration as a Process of Summation - The Calculus Primer

## The Calculus Primer (2011)

### Chapter 58. LENGTH OF A CURVE

15—8. Differential Length of Arc. It will be recalled that in §14—6 we derived a formula for the differential length of arc, namely, We shall now use the Fundamental Theorem to derive a formula for the length of an arc of a curve. By definition, the length of a portion of a curve means the limit of the sum of the chords as the number of points of division is increased indefinitely in such a manner that the length of each chord, at the same time, separately approaches zero as a limit.  Consider the length of any one of these chords, say AB, where the coordinates of A are (x′,y′), and those of B are (x′ + Δx, y′ + Δy). It will be seen that By the theorem of mean value: where x1 is the abscissa of point M on the curve at which the tangent is parallel to the chord.

Hence, AB = Δx[1 + f′(x1)2]½;

similarly,BC = Δx[1 + f′(x2)2]½,

CD = Δx[1 + f′(x3)2]½,etc.

Therefore, the length of the broken line AE is given by Thus, by the Fundamental Theorem: When using this formula, we must remember always to express in terms of x, as determined by the given equation. If y is used as the independent variable, the corresponding formula is: If the equation of the curve is given in polar coordinates, the analogous formulas for the length of an arc are  When using formula , remember to express in terms of θ before integrating; when using , the quantity must be expressed in terms of ρ before integrating.

EXAMPLE 1. Find the length of the arc of the curve x2 = 2py between the points where x = 0 and x = p.

Solution. x2 = 2py; hence . EXAMPLE 2. Find the length of the circle whose equation is x2 + y2 = r2.

Solution. Consider the quarter-arc in the first quadrant.

From the equation, . Therefore,

length of entire circle = 4 = 2πr.

EXERCISE 15—3

1. Find the length of the arc of y = x2 from the origin to the point whose abscissa is 3.

2. Find the length of the arc on the logarithmic spiral ρ = eθ from the point where θ = 0 to the point where θ = 1.

3. Find the length of the arc of y2 = 4ax between the points whose abscissas are x = 0 and x = 2a.

4. Find the length of the arc of y = log cos x between the points whose abscissas are x = 0 and x = π/6.

5. Find the length of the circle whose equation is ρ = 2a cos θ.

6. Find the length of the circle whose equation is y2 + (xa)2 = a2. Compare your result with that for Problem 5; explain.

7. Find the length of the arc on the spiral of Archimedes, ρ = aθ, when the radius vector has made one revolution, i.e., from θ = 0 to θ = 2π.

8. Find the total length of the curve whose equation is ρ = a sec θ.

9. Find the length of the arc of ev = 1 − x2 between the points whose abscissas are x = 0 and x = .

10. Find the entire length of the curve ρ = a sin3 .

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