- 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.

第十二單元

旋轉體的體積 -圓盤法

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.

Slicing the solid into thin slabs

( )V A x xD » ×D

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

Solids of Revolution

1. The Method of Disks

2. The Method of Shells

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=

Revolving about y-axis

The Method of Disks

The Method of Disks (2)

2( )dV x dyp=

V

2[ ( ) ]y dyp=

ydyp=

p= 8p==ò0

4ydyp 21

2y

0

4

Revolving about x-axis

Revolving about x-axis (animate)

The Solid by revolving about x-axis

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

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

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=

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

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-

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