Arc length
Videos, Math Dr. Bob:
- Formula for Arc Length 01: Theory and catenary
- Formula for Arc Length 02: Curve given by integral
- Formula for Arc Length 03: Reverse engineering
- Arc Length of Parabola 01: Base case
- Arc Length of Parabola 02: Sinh formula
- Arc Length of Parabola 03: Log formula
Videos, Khan Academy:
01 Theory
The total arc length of a curve is just the length of the curve.
The ‘arc length’ (not “total”) is a quantity measuring the length “as you go along,” usually given as a function of the points on the curve. It measures the length from some starting point ‘up to’ the given point.
We can use calculus to calculate the arc length of many curves. If the curve is the graph of a function, and we know the function and its derivative (whether from a formula or a data table), we can use integration to find the arc length.
Arc-length formula
The arc length
of the graph of over is: (This formula applies when
exists and is continuous on .) The arc length function
of the graph of , starting from , is:
Arc-length formula - explanation
The arc-length integral is the limit of Riemann sums that add the lengths of straight line segments whose endpoints lie on the curve, and which together approximate the curve.
Each tiny line segment is the hypotenuse of a triangle with horizontal
and vertical . We can approximate the vertical
using the derivative: Considering infinitesimals in the limit, we have
(horizontal side) and (vertical side). The Pythagorean Theorem gives: which we can simplify using
: The integral of these infinitesimals gives the arc length:
02 Illustration
Example - Arc length of chain in terms of position
12 - Arc length of chain, via position
A hanging chain describes a catenary shape. (‘Catenary’ is to hyperbolic trig as ‘sinusoid’ is to normal trig.) The graph of the hyperbolic cosine is a catenary:
Let us compute the arc length of this catenary on the portion from
to . Solution
Arc-length formula.
Give arc length
, a function of :
Compute
. Hyperbolic trig derivative:
Plug into formula.
Arc length:
Hyperbolic trig identity.
Fundamental identity:
Rearrange:
Plug into formula and compute.
Arc length:
Compute integral:
The arc length of a catenary curve matches the area under the catenary curve!
Exercise - Arc length of a line segment
13 - Arc length, line segment
Find the arc length of the straight line given by the formula
over . Check your answer using the Pythagorean Theorem.
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Surface areas of revolutions - thin bands
Videos, Math Dr. Bob
- Area of a Surface of Revolution: Derivation, cylinder, cone, sphere
03 Theory
The infinitesimal of arc length along a curve,
The general formula for the surface area is:
In any given problem we need to find the appropriate expressions for
Assuming the region is rotated around the
Area of revolution formula - thin bands
The surface area
of the surface of revolution given by is given by the formula: In this formula, we assume
and is continuous. The surface is the revolution of on around the -axis.
04 Illustration
Example - Surface area of a sphere
14 - Surface area of a sphere
Using the fact that a sphere is given by revolving a semicircle, verify the formula
for the surface area of a sphere. Solution Sphere as surface of revolution.
Sphere of radius
given by revolving upper semicircle. Upper semicircle:
Upper semicircle as function of
:
Surface area formula.
Bounds are
and . Function is
Plug data into formula:
Compute
. Power rule and chain rule:
Algebra:
Squaring:
Compute integrand.
Compute
: Integrand factors become:
Compute integral.
Surface area again:
This is the desired surface area formula
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