How to Find the Arc Length of an Archimedean Spiral: Calculus Integration Tutorial
An Archimedes spiral, also called an arithmetic spiral, is a plane curve with the property that any ray drawn from the origin and passing through the spiral will intersect the curve at equidistant points. If you look at the spiral in just one quadrant, it appears as if it is a ring of concentric and equally spaced circles, although this is actually not the case. The general form of an Archimedean spiral in polar coordinates is
r(θ) = p + qθ
where p and q are some real numbers. The parameter q determines the spacing between successive turnings of the spiral, while p determines the starting position of the spiral near the origin. For an Archimedes spiral defined by r(θ) = p + qθ, the distance between spirals is 2πq. The simplest example of an Archimedes spiral is
r(θ) = θ
where θ is measured in radians. Here, p = 0 and q = 1. The spacing between consecutive rotations is 2π. You can use the polar arc length integral to compute the length of the spiral's curve over a given interval. The general integral formula for any polar arc length is
∫ sqrt(r(θ)^2 + r'(θ)^2) dθ
and the limits of integration are the starting and ending angles of rotation. Since r(θ) = p + qθ and r'(θ) = q in this tutorial, the integral simplifies to
∫ sqrt(p^2 + q^2 + 2pqθ + (q^2)θ^2) dθ
Setting up and Evaluating the Arc Length Integral of the Spiral
For the example of r(θ) = θ, the first derivative of this function with respect to θ is r'(θ) = 1. The integral that must be evaluated is then
∫ sqrt(θ^2 + 1) dθ
In this example, we will find the length of a spiral from θ = 0 to θ = 3π, which is one and a half turns. First we must find the antiderivative of the function sqrt(θ^2 + 1). Using either a table of integrals or integration software, we find the antiderivative is
0.5θ*sqrt(θ^2 + 1) + 0.5*ln(θ + sqrt(θ^2 + 1))
Evaluating this expression at the limits of the integral gives us the arc length
0.5*3π*sqrt(9π^2 + 1) + 0.5*ln(3π + sqrt(9π^2 + 1)) - 0 - 0
For an Archimedean spiral r(θ) = p + qθ, where θ ranges from 0 to T, the exact arc length involves square roots and natural logarithms. Therefore, given an exact length L and the values of the parameters p and q, it is not possible to algebraically solve for the exact value of T.
However, there is a simpler approximation formula for the arc length L in terms of T, p, and q:
L ≈ (1/2)[ q*T^2 + (sqrt(p^2 + q^2)+p)*T ]
This approximation formula can then be solved for T in terms of p, q, and L using the quadratic formula.
More Calculus Homework Help
Here are similar articles in integral calculus to help you compute arc length and area in polar and rectangular coordinates.
- Arc Length Integral for Polar Coordinates
- Area Integral for Polar Coordinates
- Integral Calculus Tutorial: How to Compute the Arc Length of a Function y = f(x)
- How to Compute the Perimeter of an Ellipse: Integral and Approximation Formulas
- How to Find the Arc Length of an Exponential Curve: Calculus Arc Length Tutorial
- How to Find the Arc Length of a Parabolic Arc: Integral Calculus Tutorial
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