## 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′* + Δ*x, y′ +* Δ*y*). It will be seen that

By the theorem of mean value:

where *x*_{1} is the abscissa of point *M* on the curve at which the tangent is parallel to the chord.

Hence, *AB* = Δ*x*[1 + *f′*(*x*_{1})^{2}]^{½};

similarly,*BC* = Δ*x*[1 + *f′*(*x*_{2})^{2}]^{½},

*CD =* Δ*x*[1 + *f′*(*x*_{3})^{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 [3], remember to express in terms of *θ* before integrating; when using [4], the quantity must be expressed in terms of *ρ* before integrating.

EXAMPLE 1. Find the length of the arc of the curve *x*^{2} *=* 2*py* between the points where *x* = 0 and *x* = *p*.

*Solution*. *x*^{2} = 2*py*; hence .

EXAMPLE 2. Find the length of the circle whose equation is *x*^{2} + *y*^{2} = *r*^{2}.

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

From the equation, .

Therefore,

length of entire circle = 4 = 2*πr*.

**EXERCISE 15—3**

**1.** Find the length of the arc of *y* = *x*^{2} from the origin to the point whose abscissa is 3.

**2.** Find the length of the arc on the logarithmic spiral *ρ* = *e ^{θ}* from the point where

*θ =*0 to the point where

*θ*= 1.

**3.** Find the length of the arc of *y*^{2} = 4*ax* between the points whose abscissas are *x* = 0 and *x* = 2*a.*

**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* = 2*a* cos *θ*.

**6.** Find the length of the circle whose equation is *y*^{2} + (*x* − *a*)^{2} = *a*^{2}. 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* = 2*π*.

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

**9.** Find the length of the arc of *e ^{v}* = 1 −

*x*

^{2}between the points whose abscissas are

*x*= 0 and

*x*= .

**10.** Find the entire length of the curve *ρ* = *a* sin^{3} .