Area Under A Curve

D. TRAPEZOID METHOD

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1. The 4th LiveMath document (IT4AMIntegArea4.thp) shows the same speed-time graph of the car but with a series of trapeziums. There is only one way to draw the trapeziums, unlike the rectangles. In this case, n = 4. The height of each trapezium is x = (b–a)/n, like the width of each rectangle in the Riemann Sum Methods.

Question 11: Look at the diagram in the Worksheet and express the area of each trapezium PSRQ in terms of each rectangle PSRU in the Lower Riemann Sum and each rectangle TSRQ in the Upper Riemann Sum Methods.

Question 12: Hence express the sum of the trapeziums in terms of the Lower and the Upper Riemann Sums.

Question 13: By looking at the graphs in Sect B to D, which method is more accurate for the same n? The Lower Riemann Sum, the Upper Riemann Sum or the Trapezoid Method?

2. The Riemann Sum and the Trapezoid Methods are all numerical methods because they make use of the numerical values of the areas of the rectangles or the trapeziums. The answers are only approximations and the accuracy depends on the number of rectangles or trapeziums used. You will learn the calculus method later.

Continue Lesson - Section E


Question 13:	By looking at the graphs in Sect B to D, which method is more accurate for the same n? The Lower Riemann Sum, the Upper Riemann Sum or the Trapezoid Method?


2.	The Riemann Sum and the Trapezoid Methods are all numerical methods because they make use of the numerical values of the areas of the rectangles or the trapeziums. The answers are only approximations and the accuracy depends on the number of rectangles or trapeziums used. You will learn the calculus method later.