Analysis and Approaches for IBDP Maths Ebook 2 | Page 183

56
Paper 1 Section A – Volumes of Revolutions
Example
�
1
The region bounded by the curve y� cosec x and the lines x � , x � 1, y � 0 is
2
3
rotated through 2π radians about the x -axis. Find the volume of the solid generated.
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13
Solution
The volume of the solid generated
1
2
1� y dx
3
� � (A1) for correct approach
1
� � �cosec
1
3
2
�
d
2 xx
�
Let u � x. (M1) for substitution
2
du
�
� � 2du�
�dx
dx
2
� �
x�1 � u� (1) �
2 2
1 � �1�
�
x� � u � � ��
3 2 �3 � 6
�
�
1
2 2 2
1�
y dx � � 2cosec udu
3 6
� � (A2) for correct working
�
1
2 2
1�
y dx ���2cot
u��
3 6
� A1
�
�
1
1
3
2 � � � �
� y dx� �2cot � � �2cot
�
2 � 6�
1
2
1� y x� � � �
3
d 2(0) ( 2 3)
1
2
� 1�
y dx�
2 3
A1
3
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