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Trapezium Rule

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Published in: Mathematics
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The Trapezium Rule (or Trapezoidal Rule) is a numerical method used to approximate the definite integral of a function. This document provides a visual illustration of the method plus an illustrative example.

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  1. The trapezium rule The trapezium rule provides you with a way to estimate the value of an integral you cannot do. It involves splitting the area under the curve up into trapeziums which are then totaled to give an estimate for the area. 12 If we wanted to find the region between the curve and x = 7 and x = 12 we would split it up into equal width strips, with more strips leading to a more accurate result. We'd then find the y values for the x values of our strips from either the graph or the equation if we are given it. We then use the trapezium rule equation to approximate y dx Where n is the number of strips and h is the width of the strips = computed from the function by plugging x= a, a+h, a+2h,... , n , yi can be Example question: Use the trapezium rule with 4 strips to estimate the area enclosed by the curve of (l c 4 and the lines x = 2 and x = 6 Worked solution: For 4 strips between x = 2 and x = 6 each strip needs to have a width of l, so our x values are 2, 3, 4, 5 and 6 (note that there aren + I x values). It helps to draw up a table to calculate the corresponding y values from the equation:
  2. x 2 3 4 5 6 -0.53 -0.094 -0.031 -0.014 0.0074 The minus numbers here simply indicate that the area is below the x-axis. We can now estimate the area: -0.8154 Z 0.5374 — = 2 Note that: 1) If the curve is bending "outwards", then the trapezium rule approximation method will be an "underestimate" (because Of the small wasted areas from the curve that are not taken into consideration) 2) If the curve is bending "inwards", then the trapezium rule approximation method will be an "overestimate" 0.031+— 0.4077 l) 0.014)]

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