Apps. and Interpretation for IBDP Maths Ebook 1 | Page 225

64
Paper 1 – Errors in Trapezoidal Rule
Example
Consider the integral
�
0
6
2
(1 � x)
2
dx
. The value of this integral is estimated by trapezoidal
rule with 6 trapeziums, such that the width of each trapezium is 1.
(a) Find an estimate of
It is given that the exact value of
2
(1 � x)
6
� dx
.
0 2
2
(1 � x)
6
� dx
is 12
0 2
7 .
[3]
16
(b)
Calculate the percentage error in the estimate.
[2]
Solution
(a)
(b)
The estimate of
�
0
6
2
(1 � x)
2
dx
� 2 2
�
�
�
2 2
(1 �0) (1 �6)
�
�
�
1 � � 2 2
��
� (1) � 2 2
2
� �
(1 1) (1 2)
��
�
� �
�2�
��
� � 2 2 2 ��
� �� � �
2 2 2 �
(1 � 3) (1 � 4) (1 � 5)
�
� �
��
(A2) for substitution
� 2.003185941
� 2.00
A1 N3
[3]
The percentage error
12
2.003185941�
� 7 � 100%
(A1) for correct substitution
12
7
� 16.85251323%
� 16.9%
A1 N2
[2]
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