Quadratic Segments

The investigation begins by looking at the area of three segments related to a quadratic. The first segment is the area between the curve and the x-axis. Each of the other segments are determined by the difference between the area of a trapezium formed by joining two points on the curve and the x-axis, and then subtracting the area under the curve.

Finding a result for the general case involves finding the area of a trapezium and using integration to find the area under the curve. The final result is dependent upon the quadratic coefficient and the width of the trapezium. The fact that the linear coefficient has no effect is not surprising, but the independence in relation to the linear coefficient is a surprise to most. This makes for a nice link with the work of the ‘Something in Common’ challenge ‘Parabola in Parallelogram’.