Volume by Integration, Lecture notes of Engineering Mathematics

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APPLICATIONS

VOLUME BY INTEGRATION

A solid of revolution is the figure formed when a plane region is revolved

about a fixed line. The fixed line is called the axis of revolution. For short,

we shall refer to the fixed line as axis.

The volume of a solid of revolution may be using the following

methods:

DISK, RING and SHELL METHOD

DISK METHOD: V = r

2

h

Find the volume of the solid generated by revolving the region bounded by the

line y = 6 – 2x and the coordinate axes about the y-axis.

r =x

( x , y )

h = dy

( 0 , 6 )

x

y

0

(3,0)

𝑉 =



0

6

𝜋 𝑥

2

𝑑𝑦

𝑉 =

0

6

𝜋

[

1

2

]

2

𝑑𝑦

𝑉 =

𝜋

4



0

6

( 6 − 𝑦 )

2

𝑑𝑦

( 6 − 𝑦 )

3

|

0

3

𝑥=

1

2

( 6 − 𝑦

)

𝑖𝑓 𝑦 = 6 − 2 x ;

𝑉 = −

𝜋

12

[

( 6 − 6

)

3

−

( 6 − 0

)

3

]

V r h

2

 

Ring or Washer method is used when the element (or representative strip) is perpendicular to

but not touching the axis. Since the axis is not a part of the boundary of the plane area, the strip

when revolved about the axis generates a ring or washer.

B. RING OR WASHER METHOD: V = (R

2

• r

2

)h

( x

1

, y

1

)

( x

2

, y

2

)

x = a

x = b

dx

h = dx

y

1

= g(x)

y

2

= f ( x )



❑

❑

𝑑𝑉 =



❑

❑

𝜋

[

𝑦

1

2

− 𝑦

2

2

]

𝑑𝑥

Since

𝑦

1

= 𝑓 (𝑥)

𝑦

2

=𝑔 (𝑥)

𝑉 =𝜋



𝑎

𝑏

[ 𝑔 ( 𝑥 )

2

− 𝑓 ( 𝑥 )

2

] 𝑑𝑥

r

R

C. SHELL METHOD:

The method is used when the

element (or representative strip) is

parallel to the axis of revolution.

When this strip is revolved about

the axis, the solid formed is of

hollow cylindrical form.

𝑉

𝑠 𝑒𝑙𝑙 h

= 2 𝜋 𝑟𝑡 ⋅ h

SHELL METHOD:

Find the volume of the solid generated by revolving the second quadrant region bounded

by the curve about the line.

𝒙

𝟐

= 𝟒 − 𝒚

𝑑𝑉 = 2 𝜋 ( 1 − 𝑥 ) 𝑦𝑑𝑥

𝑉 = 2 𝜋 𝑟 𝑡 h

𝑉 = 2 𝜋



− 2

0

( 1 − 𝑥 ) 𝑦 𝑑𝑥



0

− 2



0

− 2

but x

2

= 4 − 𝑦 ; y =4− x

2



0

− 2

[

( 4 − 𝑥

2

) − 𝑥

( 4 − 𝑥

2

) ] 𝑑𝑥



0

− 2

( 4 − 4 𝑥 − 𝑥

2

3

) 𝑑𝑥

𝑉 =

56 𝜋

3

𝑐𝑢. 𝑢𝑛𝑖𝑡𝑠