length of cures,surface area, volume of revolution
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8/9/2019 Length of Cures,Surface Area, Volume of Revolution
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8/9/2019 Length of Cures,Surface Area, Volume of Revolution
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1. In rectangular form(y=f(x) or x=g(y) ), use whichever of the following is
easier
i)
ii)
2. In parametric form ( x=x(t), y=y(t) )
3. In Polar form
8/9/2019 Length of Cures,Surface Area, Volume of Revolution
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Surface Areas Of RevolutionAny Curve AB when revolves about any line L, the surface areagenerated is
Where is the perpendicular distance
from any point on the curve AB to the
axis of revolution L.
The curve is restricted between a & b.
a b L
Y
A
B
X
8/9/2019 Length of Cures,Surface Area, Volume of Revolution
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(a) Cartesian form
(i) Revolution about x-axis
Any Curve AB when revolves about X-axis, the surface area
generated is
Where y is the perpendicular distance
from any point on the curve AB to the
axis of revolution X.The curve is restricted between a & b.
a b X
Y
y
A
B
X
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ii) Revolution about y-axis
Any Curve AB when revolves about Y-axis, the surface areagenerated is
Where x is the perpendicular distance
from any point on the curve AB to the
axis of revolution Y.
The curve is restricted between a & b.
Y
X
x
B
A
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(b) Parametric form
(i) Revolution about x-axis
Any Curve AB when revolves about X-axis, the surface area
generated is
Where y is the perpendicular distance
from any point on the curve AB to the
axis of revolution X.The curve is restricted between a & b.
a b X
Y
y
A
B
X
8/9/2019 Length of Cures,Surface Area, Volume of Revolution
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(ii) Revolution about y-axis
Any Curve AB when revolves about y-axis,
the surface area generated is
Where x is the perpendicular distance
from any point on the curve AB to the
axis of revolution x.The curve is restricted between a & b.
Y
a
b
X
x
A
B
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(C) Polar form
(i)Revolution about =0 line
Any Curve AB when revolves about X-axis, the surface area
generated is
Where y()=r Sin.
The curve is restricted between a & b.
X
=0
= 1
= 2
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(C) Polar form
(i)Revolution about line
Any Curve AB ( )when revolves about line, the surface area
generated is
Where x()=r Cos.
The curve is restricted between a & b.
X
=0
= 1
= 2
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8/9/2019 Length of Cures,Surface Area, Volume of Revolution
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i) Revolution about x-axis
When area under the curve AB revolves about x-axis, the volumegenerated is
Where y is the perpendicular distance
from any point on the curve AB to the
axis of revolution X.
The curve is restricted between a & b.
a b X
Y
A
B
Y
8/9/2019 Length of Cures,Surface Area, Volume of Revolution
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ii) Revolution about y-axis
When area under the curve AB revolves about Y-axis, the volumegenerated is
Where x is the perpendicular distance
from any point on the curve AB to the
axis of revolution Y.
The curve is restricted between a & b along y-axis.
Y
X
x
B
A
x
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(C) Polar form
(i)Revolution about =0 line
Any Curve AB(r=f()) when revolves about =0 line, the volume
generated is
=0
= 1
= 2
A
B
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(C) Polar form
(i)Revolution about line
Any Curve AB ( )when revolves about line, the surface area
generated is
X
=0
= 1
= 2
A
B