Analysis and Approaches for IBDP Maths Ebook 1 | Seite 197

67
Paper 1 Section A – Area under a curve
Front Page
Example
Solution
2
x
Consider the graph of f( x)
�
3
x � 1
, where f( x) � 0 for 0 � x � 2 . Let R be the region
enclosed by the graph of f , the x -axis and the line x � 2 . Find the area of R .
[6]
The area of R
2
� � f ( x)dx
(A1) for definite integral
0
2
2 x
� � dx
0 3
x �1
3
Let u�x
� 1
3
x� 2 � u� 2 �1�
9
du
2
3x
dx 3
x� 0 � u� 0 �1�
1
1 d 2 3 u x x
9 1 1
�� � d
1
u 3 u
(A2) for substitution
1 9 1
� d u
3�
1
u
1
� � ln u� 9 1
3
A1
1
� (ln 9 � ln1)
3
(M1) for substitution
1
� ln 9
3
A1 N3
[6]
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