Length of a Spiral
What about spirals, which are expressed in polar coordinates, where r, the distance from the origin, is a function of the theta (θ) angle that's made with the x axis? We can't use our x and y functions to measure the spiral shown in the following diagram:
Figure 11.10: An Archimedean spiral
What we have in the preceding diagram is a spiral that starts at (5,0) and makes 7.5 turns, ending at (11,π). The formula for that curve is r(θ) = 5 + 0.12892θ. The number of radians turned is 7.5 times 2π, which is 15π. We're going to use the same idea as in the previous section: we're going to find the length of the straight line from r(θ) to r(θ+step) for some tiny step in the central angle, as shown in the following diagram:
Figure 11.11: Approximating the length of a tiny part of the curve
The opposite side to the central angle of the triangle shown in the...
