Find the Total Area between the curve x^3 -3x^2 +2x and the x-axis, when 0 ≤ x ≤ 2.

Initally this question appears to be a straightforward integration that should be carried out between the limits of x=0 and x=2. However if this process is carried out, as strange result is obtained: Area = 0? A student might be conviced by their working and leave this result as their answer. Howvever this is incorrect as it has been found by calculating the sum of areas both above and below the x-axis. This can be visualised more easily when a sketch of the graph is drawn. From the question, it can be clearly seen that the curve is a positive cubic (indicating the overall shape), however to sketch this curve without a calculator, the cubic can be easily factorised to x(x^2 -3x+2), indicating a root at x=0. The quadrating component can then be solved using completing the square or the quadratic formula to obtain the remaining roots, x=1 and x=2. When the curve is now sketched, it can be seen much more clearly why the initial result was obtained. This is because the positive area beteeen the curve and the x-axis for 0≤x≤1 is equal in magnitude to the 'negative' area for when 1≤x≤2. This observation can be used to conclude that the Total Area is equal to twice the magnitude of one of these sections. Standard integration then gives the result: Total Area = 2*1/4 = 1/2