The Calculus Primer (2011)
Part XV. Integration as a Process of Summation
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, .
length of entire circle = 4 = 2πr.
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 + (x − a)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 .