Multivariate Calculus - Bruce E. Shapiro

Lecture 5
Velocity, Acceleration, and
Curvature
Length of a Curve
Our definition of the length of a curve follows our intuition. Lay a string along the
path of the curve, pick up the string, straighten it out, and measure the length of
the line.
Theorem 5.1 Suppose that a curve is parameterized as
r(t) = (f(t), g(t), h(t)) (5.1)
on some interval at < b. Then the arc length from a to b or the length of the
curve from a to b is given by the integral
s =
∫ b
a
∣
∣r ′ (t) ∣ dt =
∫ b
a
√
(f ′ (t)) 2 + (g ′ (t)) 2 + (h ′ (t)) 2 dt (5.2)
Proof. Let n be some large integer and define ɛ = (b − a)/n. Divide [a, b] into n
intervals
[a, a + ɛ], [a + ɛ, a + 2ɛ], ..., [b − ɛ, b]
and approximate the curve by straight line segments
r(a)r(a + ɛ), r(a + ɛ)r(a + 2ɛ), ..., r(b − 2ɛ)r(b − ɛ), r(b − ɛ)r(a − ɛ),
The end points of the path during the time interval from t ∈ [a + iɛ, a + (i + 1)ɛ]
are given by
r(a + iɛ) = (f(a + iɛ), g(a + iɛ), h(a + iɛ))
and
r(a + (i + 1)ɛ) = (f(a + (i + 1)ɛ), g(a + (i + 1)ɛ), h(a + (i + 1)ɛ))
Hence the length of the line segment from r(a + iɛ) to r(a + (i + 1)ɛ)
37