The filled red point [math]\left(h,y_2\right)[/math] can be moved to positive integer values of [math]x[/math] and integer values of [math]y[/math]. The unfilled point [math]\left(0,y_1\right)[/math] is restricted to the [math]y[/math]-axis but can be moved up and down. The unfilled point [math]\left(-h,y_0\right)[/math] is dependent on [math]\left(h,y_2\right)[/math] but can be moved up and down. [b]Task 1:[/b] Show that the equation of the parabola is [math]\displaystyle y=\left( \frac{y_0-2y_1+y_2}{2h^2}\right)x^2+\left( \frac{y_2-y_0}{2h}\right)x+y_1[/math]. [b]Task 2:[/b] Show that [math]\displaystyle \int_{-h}^h \left( \frac{y_0-2y_1+y_2}{2h^2}\right)x^2+\left( \frac{y_2-y_0}{2h}\right)x+y_1 \mbox{ d}x = \frac{h}{3} \left( y_0+4y_1+y_2 \right) [/math]. [b]Task 3:[/b] If all three points are translated so that [math]\left(-h,y_0\right) \rightarrow \left(-h+\alpha ,y_0\right)[/math], [math]\left(0,y_1\right) \rightarrow \left(\alpha ,y_1\right)[/math] and [math]\left(h ,y_2\right) \rightarrow \left(h+\alpha ,y_1\right)[/math], what happens to the area under the curve?
Created by Dr GJ Daniels.