SUMMARY POINTs
11
� Areas on x - y plane , between y� c and y� d : d 1 . � g( y) dy : Area between the graph of g( y ) and the y -axis c d
2 . � g( y) � f ( y) dy : Area between the graph of g( y ) and the graph of c f( y )
� Volumes of revolutions about the x -axis, between x� a and x� b:
1 .
2 . b
2
( ( )) d a
V � �� f x x : Volume of revolution when the region between the
graph of f( x ) and the x -axis is rotated 360 about the x -axis b
2 2
(( ( )) �( ( )) ) d a
�
V �� f x g x x : Volume of revolution when the region between the graphs of f( x ) and gx ( ) is rotated 360 about the x -axis
� Volumes of revolutions about the y -axis, between y� c and y� d : 1 .
2 . d
2
( ( )) d c
V � �� g y y : Volume of revolution when the region between the
graph of g( y ) and the y -axis is rotated 360 about the y -axis d
2 2
(( ( )) �( ( )) ) d c
�
V �� g y f y y : Volume of revolution when the region between the graphs of g( y ) and f( y ) is rotated 360 about the y -axis
� Applications in kinematics : 1 . at (): Acceleration with respect to the time t
2 . |
v( t) � � a( t) dt
: Velocity
|
3 . |
s( t) � � v( t) dt
: Displacement
|
4 . t2 d � � v( t) dt : Total distance travelled between t 1
and t 2 t1
Solutions of Chapter 11
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