The trapezium rule, with 2 intervals of equal width, is to be used to find an approximate value for fe"dx . i) Explain with the aid of a sketch, why the approximation will be greater than the exact value of the integral. ii) Calculate the approximate value and the exact value, giving each answer correct to 3 decimal places. ii1) Another approximation to 'e*dx is to be calculated by using two trapezium of unequal width. The first trapezium has width h and the second trapezium has width (1– k), so that the three ordinates are at x= 0, x=h and x=1. Show that the total area T of these two trapezium is given by 7 =(r! + h(1- e*1) + e* ). iv) Show that the value of h for which T is a minimum is given by h = In'
The trapezium rule, with 2 intervals of equal width, is to be used to find an approximate value for fe"dx . i) Explain with the aid of a sketch, why the approximation will be greater than the exact value of the integral. ii) Calculate the approximate value and the exact value, giving each answer correct to 3 decimal places. ii1) Another approximation to 'e*dx is to be calculated by using two trapezium of unequal width. The first trapezium has width h and the second trapezium has width (1– k), so that the three ordinates are at x= 0, x=h and x=1. Show that the total area T of these two trapezium is given by 7 =(r! + h(1- e*1) + e* ). iv) Show that the value of h for which T is a minimum is given by h = In'
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:b)
The trapezium rule, with 2 intervals of equal width, is to be used to find an approximate
value for fe"dx .
i)
Explain with the aid of a sketch, why the approximation will be greater than the
exact value of the integral.
ii)
Calculate the approximate value and the exact value, giving each answer correct to
3 decimal places.
i11)
Another approximation to
'e*dx is to be calculated by using two trapezium of
...
unequal width. The first trapezium has width h and the second trapezium has width
(1– k), so that the three ordinates are at x= 0, x=h and x=1. Show that the total
area T of these two trapezium is given by T =(e-1 + h(1- e-) + ei ).
(e)
Show that the value of h for which T is a minimum is given by h = In
iv)
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