Apps. and Interpretation for IBDP Maths Ebook 1 | Page 221

62
Paper 1 – Integration and 3-D Objects
Example
Consider the graph of f ( x) � 3( x �1)(5 � x)
, where f( x) � 0 for 1�x
� 5. Let R be the
region enclosed by the graph of f and the x -axis.
(a) (i) Write down the integral representing the area of R .
(ii) Hence, find the area of R .
A triangular prism is formed with height 7.5 and the cross-sectional area equals to the
area of R .
[2]
16
(b)
Find the volume of the triangular prism.
[2]
Solution
(a)
(i)
1
5
� 3( x � 1)(5 �x
)d x
A1 N1
(b)
(ii) 32 A1 N1
The volume of the triangular prism
� (32)(7.5)
(A1) for substitution
� 240
A1 N2
[2]
[2]
Exercise 62
1. Consider the graph of
f ( x) 1.5( x 2)( x 10)
2
� � � , where ( ) 0
be the region enclosed by the graph of f and the x -axis.
f x � for �2 � x � 10 . Let R
(a) (i) Write down the integral representing the area of R .
(ii) Hence, find the exact area of R .
[2]
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