The integral cannot be expressed in fnite terms Your mission

The integral

cannot be expressed in fnite terms.

Your mission is to show that it gives the length of a certain figure-8 shaped curve.

The curve can be described as the solution set of the equation

is a parametrization of part of the curve, that is, has image contained in the solution set of

Write the ingredients for the arc length integral and show that it is the integral given.

Solution

Place the slower boat initially at the origin of a coordinate plane, and equate \"north\" with the positive y-axis, and east with the positive x-axis. The faster boat starts at the point (50,0). The \"curve\" shall refer to the path of the faster boat. t shall refer to time since both boats left their starting positions.

There are two ways to indicate the slope at any point along the curve. One is dy/dx. The other is (x-20t)/y, since the faster boat is always pointing towards the slower boat. Thus dy/dx=y/(x-20t).

The above equation can also be derived this way (optional):

Pick a point on the curve and label it (x₁,y₁). The x₁ and the y₁ are functions of the current time, t₁. The tangent to the curve at point (x₁,y₁) passes through the point (20t₁,0), which is the current location of the 20kph boat. The equation of the line is y=mx+b, where m is the value of dy/dx at (x₁,y₁) and b is some constant chosen to make sure that (20_t1,0) is on the line. b must be -20*t₁*y\'(x₁), where I\'ve used y\'(x₁) to mean the value of dy/dx evaluated at the point (x₁,y₁). The equation of the tangent line is therefore:

y = y\'(x₁)*x - 20*t₁*y\'(x₁)

Rearranging that, and phrasing it as a general statement rather than a particular point (the point x₁ was arbitrary, so this applies to every point on the curve).

dy/dx * (x-20t) = y
dy/dx = y/(x-20t)