Question

Task 2 - Trapezium Rule (Maximum Mark 10) Find the equation of the line through the points A(Y) and (.ya) and find an expression for the area under this line between the points A and B. Explain carefully how your result can be used to prove the general formula for the Trapezium Rule. Notes: • This part of the assignment is testing that you can find the equation of a line and use this to derive other formulae. Marking Criteria Topic Equation of line and area Poor Good 0 - 1 mark 2 - 3 marks No explanation, just finds the equation quotes formulae. of the line and the area, with basic explanation Only states formula, Basic derivation of basic explanation of formula and terms. connection to name, tenuous connection to previous result. Excellent 4-5 marks Finds the equation of the line and the area, with full explanation. Detailed explanation Derivation of Trapezium Rule why is it called the trapezium rule connection to previous result

Answer 1

This result can be used to prove the general formula for the trapezium rule.
Assume that we have a trapezium as given below :


where h is the height of the trapezium.

Now break the trapezium into two parts (one above the x-axis and one below the x-axis) :


where c+e = a and d+f = b.


The area of the trapezium above the x-axis is h(c+d)/2
The area of the trapezium below the x-axis is h(e+f)/2

Thus, the area of the original trapezium = h(c+d)/2 + h(a+b)/2 = h( (c+e) + (d+f) )/2 = h(a+b)/2.
This proves the Trapezium Rule.