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The Trapeziodal Rule

THE TRAPEZOIDAL RULE

Trapezoidal Rule uses the first order Taylor series to approximate f(x).

Then

So trapezoidal rule is equivalent to computing the area under the linear polynomial passing through points a and b.

Figure 1: Deriving the Trapezoidal Formula

Source: (RAHAM, 2003).

In multiple segment Trapezoidal rule the area is divided into equal segments and the width of each segment is

The Integral I is, which can be broken into h integrals as,

(Hairer & Wanner, 2010)

The true error in each segment is given by:

Hence the total error in multiple segment Trapezoidal rule is

Question 1

A company advertises on OLX that every roll of toilet paper has at least 250 sheets but not exceeding 270 sheets. The probability that there are 250 or more sheets in the toilet paper is given by

Use single segment Trapezoidal rule to find the probability that there are 250 or more sheets.

Solution

, where

Question 2

In an attempt to understand the mechanism of the depolarization process in a fuel cell, an electro-kinetic model for mixed oxygen-methanol current on platinum was developed in the laboratory at Kenya Industrial Research & Training Institute (KIRDI). A very simplified model of the reaction developed suggests a functional relation in an integral form. To find the time required for 50% of the oxygen to be consumed, the time, is given by

Use two- segment Trapezoidal rule to find the time required for 50% of the oxygen to be consumed.
Find the true error, , for part (a).
Find the absolute relative true error, , for part (a).

Solution

b) The exact value of the above integral is,

so the true error is

c) The absolute relative true error, , would then be

Values obtained using multiple-segment Trapezoidal rule for the equation can be tabulated as:

	Value
1	191190	-1056.2	0.55549	—
2	190420	-282.12	0.14838	0.40711
3	190260	-127.31	0.066956	0.081424
4	190210	-72.017	0.037877	0.029079
5	190180	-46.216	0.024307	0.013570
6	190170	-32.142	0.016905	0.0074020
7	190160	-23.636	0.012431	0.0044740
8	190150	-18.107	0.0095231	0.0029079

References

RAHAM, A. (2003). Statistics . Blacklick, Ohio, McGraw-Hill.
HAIRER, E., & WANNER, G. (2010). Solving ordinary differential equations II. Berlin, Springer. http://public.eblib.com/EBLPublic/PublicView.do?ptiID=571353.

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