Volumes by Cylindrical Shells

r

Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä

Cylindrical Shells

The method applies to solids obtained by letting the domain under the graph of a function rotate about the vertical axis.

Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä

Cylindres

The volume of a cylinder of height h on a disk of radius R is

V = height ⨉ area of the bottom

V = πR2h

h

R

Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä

Cylindrical Shells

A Cylindrical Shell is obtained by removing, from a solid cylinder, a

cylinder of smaller radius as indicted in the

picture.

h

rRr

Volume of the Cylindrical Shell = πR2h - πr2h = πh (R2 - r2) = πh (R - r)(R + r).

Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä

Cylindrical Shells

h

rRr

Area of the Rectanlge A = (R - r)h.

Volume of the Shell= πh(R - r)(R + r) = .

h

Rr

A Cylindrical Shell can also be obtained by

letting the green rectangle in the picture rotate about the vertical

axis.

2π R + r

2A

Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä

Cylindrical Shells

When all the rectangles of a Riemann sum rotate about the

vertical axis we get a Cylindrical Shell Approximation

of the Volume of the solid obtained by letting

the domain bounded by the graph of the given function rotate about the vertical axis.

Volume of the Shell= . 2π R + r

2A

Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä

Cylindrical Shell Approximations

f

Let xk be the midpoint of a subinterval. Rectangle of height f(xk) and width Δx has the area f(xk) Δx.

The rectangle rotates about the vertical axis forming a cylindrical shell of volume Vk = 2πxk f(xk) Δx

assuming that xk > 0.

Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä

Cylindrical Shells

When all the rectangles of a Riemann sum rotate about the vertical axis we get a

Cylindrical Shell Approximation of the Volume.

V

D= 2πx

kf x

k( ) Δxk=1

n

∑

D → 0⏐ →⏐ ⏐ ⏐ 2πx f x( )dx

a

b

∫ =V

Valid if 0 ≤ a ≤ b and if f is non-negative on the interval [a,b].

Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä

Cylindrical Shells Formula

The Volume of a Solid obtained by letting the domain bounded by the

graph of a function f rotate about the vertical axis:

V = 2π x f x( ) dx

a

b

∫Here we assume the (a,b) is either an interval in the positive real axis or in the negative real

axis. I.e. we assume that 0 ∉ (a,b).

Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä

Example

V = 2πx −9 +12 x −3x2

( )dx1

3

∫ f x( ) =−9 +12 x −3x2

1,3⎡⎣

⎤⎦

Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä

Example

V = 2πx −9 +12 x −3x2

( )dx1

3

∫

=2π −9 x +12 x2 −3x3

( )dx1

3

∫

=16π =2π −

9 x2

2+ 4 x3 −

3x4

4

⎡

⎣⎢

⎤

⎦⎥1

3

V = 2π x f x( ) dx

a

b

∫ f x( ) =−9 +12 x −3x2

Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä

Cylindrical Shells Formula

V = 2π x f x( ) dx

a

b

∫

The Volume of a Solid obtained by letting the domain bounded by the

graph of a function f rotate about the vertical axis:

Here we assume the (a,b) is either an interval in the positive real axis or in the negative real

axis. I.e. we assume that 0 ∉ (a,b).