Chapter 5: Applications of Integration
Section 5.4: Arc Length
Essentials
Table 5.4.1 lists arc-length formulas for curves given explicitly, and parametrically. The arc-length function s, defined by an appropriate integral, gives the length of a curve from some initial point to a varying endpoint. The differential of this function is ds, the arc-length element.
Curve
Arc-Length Element (ds)
Arc-Length Function
y=fx
ds=1+dfdx2 dx
sx^=∫ax^1+dfdx2 ⅆx
x=gy
ds=1+dgdy2 dy
sy^=∫ay^1+dgdy2 ⅆy
x=xt
y=yt
ds=dxdt2+dydt2dt
st^=∫at^dxdt2+dydt2 ⅆt
Table 5.4.1 Arc-length formulas
When a limit of integration in a definite integral is a variable, that variable must not coincide with the independent variable in the integrand. That is why x^, y^, and t^ are used in the definitions of the arc-length functions.
Because of the square root appearing in arc-length integrals, these integrals are often extremely difficult, it not impossible, to evaluate in terms of elementary functions.
A word on grammar: When used as a noun, "arc length" is typically not hyphenated. However, when used as an adjective, as in "arc-length function", this text employs the hyphen.
Examples
Example 5.4.1
For the curve defined by y=x3/2,0≤x≤4, obtain the length, and the arc-length function on 0≤x^≤4.
Example 5.4.2
For the curve defined by x=y2, obtain the length of the arc from 0,0 to 1,1.
Example 5.4.3
Obtain the length of the curve defined by y=sinx,0≤x≤ π.
Example 5.4.4
Obtain the length of the curve defined by y=sinx2,0≤x≤2.
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