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第十二單元
旋轉體的體積 -圓盤法
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Volumes of a Solid The volumes of solid that can be cut into thin slices,
where the volumes can be interpreted as a definite integral.
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Slicing the solid into thin slabs
( )V A x xD » ×D
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The formula of volume of Solid
The volume V of the solid should be given approximately by the Riemann sum
1
( )n
i ii
V A x x=
» Då
( )b
aV A x dx=ò
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Solids of Revolution
1. The Method of Disks
2. The Method of Shells
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Example Let R be the regions bounded by the graphs of
and Compute the volume of the solid formed by revolving R about y-axis
2 0,y x x= = 4y=
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Revolving about y-axis
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The Method of Disks
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The Method of Disks (2)
2( )dV x dyp=
V
2[ ( ) ]y dyp=
ydyp=
p= 8p==ò0
4ydyp 21
2y
0
4
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Revolving about x-axis
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Revolving about x-axis (animate)
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The Solid by revolving about x-axis
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The Solid by revolving about x-axis
2 24dV y dxp pé ù= -ê úë û
V =
2 216 ( )x dxp pé ù= -ê úë û416 x dxp pé ù= -ê úë û
5116
5x xp p= -
32 12832
5 5p p p= - =
ò0
2416 x dxp pé ù-ê úë û
0
2
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Example Let R be the regions bounded by the graphs of
and Compute the volume of the solid formed by revolving R about y-axis
2y x=
x
y
y x=
2y x=y x=
(1,1)
dy
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Example(continued)
x y= x y=dy2 2[ ( ) ( ) ]dV dyyyp p= -
2[ ]y dyyp p= -
V =ò 2[ ]y dyyp p-0
1
21
2yp= × 31
3yp- ×
0
1
2
p=
3
p-
6
p=
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Example Let R be the regions bounded by the graphs of
and Compute the volume of the solid formed by revolving R about x-axis
2y x=
x
y
y x=
2y x=y x=
(1,1)
dx
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Example(continued)
22 2[ ( ) ( ) ]dV xxx dp p= -
2 4[ ]x dxxp p= -
V =ò0
1
31
3yp= × 51
5yp- ×
0
1
3
p=
5
p-
2
15
p=
y x=
2y x=dx
42[ ] dxxxp p-
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單元結語
求旋轉體的體積不要死背公式,要用分割切薄片的觀念解題,公式只要遇到變化的狀況就無效了。
一般來說,對付簡單的題型,圓盤法計算及觀念較簡單,適宜初學者使用