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.

Make sure you have derived the results from 'Towards Simpson's Rule'. [url]http://www.geogebratube.org/student/m56927[/url] The blue dots represent [math]\left \{ y_0,y_2,y_4,\ldots,y_n\right \}[/math]. The pink dots represent [math]\left \{ y_1,y_3,y_5,\ldots,y_{n-1}\right \}[/math]. You can change the function, move the red dots to change the upper and lower limits of the integration. You can also change the number of intervals. For clarity, only the first two parabolic approximations are shown. [b]Task 4:[/b] See how the increasing the number of intervals increases the accuracy. Can you find an integral where the accuracy isn't very good? [b]Task 5:[/b] Show that [math]\displaystyle \int_a^b \mbox{f}(x) \mbox{ d}x \approx \frac h 3 \left[ y_0 + y_n + 4\left( y_1+y_3 +\ldots +y_{n-1}\right)+2\left( y_2+y_4 +\ldots +y_{n-2}\right)\right] [/math].

Created by Dr GJ Daniels.