Solutions to Exercises

Chapter 2

Example 2.2 – Double wiper overlap problem

The area of the curved region ABCD is required.

Taking Oxy axes as shown, then the area is formulated in terms of the double integral in polar coordinates:

The principle difficulty in evaluating this is the upper limit BC which is the polar equation of the curve BC. First, the equation of the circle containing BC in cartesian coordinates is:

Transforming into ‘polars’ by substituting x = r cos θ and y = r sin θ and manipulating to get r explicitly gives:

The integral looks a little unpleasant, but if a computer algebra package such as DERIVE is used, then the answer is immediately delivered:

Trial data: r = 40 cm, L = 10 cm, w = 20 cm, α = 2π/3 gives Area ABCD = 1578 cm 2 .