Solid of Revolution

Revolution about x-axis

What is a Solid of Revolution - 1

Consider the area under the graph of y = 0.5x from x = 0 to x = 1:

What is a Solid of Revolution - 2

If the shaded area is now rotated about the x-axis, then a three-dimensional solid (called Solid of Revolution) will be formed:

http://chuwm2.tripod.com/revolution/Pictures from

What will it look like?

What is a Solid of Revolution - 3 Actually, if the shaded triangle is regarded as made up of straight lines perpendicular to the x-axis, then each of them will give a circular plate when rotated about the x-axis.  The collection of all such plates then pile up to form the solid of revolution, which is a cone in this case.

Finding Volume

http://clem.mscd.edu/~talmanl/HTML/VolumeOfRevolution.html

How is it calculated - 1Consider the solid of revolution formed by the graph of y = x2 from x = 0 to x = 2:

What will it look like?

http://www.worldofgramophones.com/victor-victrola-gramophone-II.jpg

How is it calculated - 2Just like the area under a continuous curve can be

approximated by a series of narrow rectangles, the volume of a solid of revolution can be approximated by a series of thin circular discs:

we could improve our accuracy by using a larger and larger number of circular discs,making them thinner and thinner

How is it calculated - 3

x

x

x

As n tends to infinity,It means the discs get thinner and thinner.

And it becomes a better and better approximation.

As n tends to infinity,It means the discs get thinner and thinner.

And it becomes a better and better approximation.It can be replaced by an integral

Volume of Revolution Formula

The volume of revolution about the x-axis between x = a and x = b, as , is :

This formula you do need to know

Think of is as the um of lots of circles

… where area of circle = r2

Δx = 0

limΔx→ 0

π f x( )2⎡⎣

⎤⎦

x=a

x=b

∑ Δx= π f x( )2⎡⎣

⎤⎦a

b

dx

0

1

2

1 2 3 4

y = x How could we find the volume of the cone?

One way would be to cut it into a series of disks

(flat circular cylinders) and add their volumes.

The volume of each disk is:2 the thicknessrπ ⋅

In this case:

r= the y value of the function

thickness = a small change

in x = dx

π π x( )

2

dx

→

Example of a disk

0

1

2

1 2 3 4

y = x

The volume of each flat cylinder (disk) is:

2 the thicknessrπ ⋅

If we add the volumes, we get:

( )24

0x dxπ∫

4

0 x dxπ=

42

02x

π= 8π=

π x( )

2

dx

Example 1

dxx1

0

2)5.0(

Consider the area under the graph of y = 0.5x from x = 0 to x = 1:

What is the volume of revolution about the x-axis?

Integrating and substituting gives:

0.5 1

π y2

a

b

∫ dx π (0.5x)2

0

1

∫ dx = π

4x2

0

1

∫ dx

π4

x3

3

⎡

⎣⎢

⎤

⎦⎥

0

1

=π

4

1

3− 0⎡

⎣⎢⎤⎦⎥

=π

12

Example 2

between x = 1 and x = 4

What is the volume of revolution about the x-axis

Integrating gives:

x

x1y

+=for

π y2

a

b

∫ dx π1+ x

x

⎛

⎝⎜⎞

⎠⎟

2

1

4

∫ dx  = π1+ x

x1

4

∫ dx

=π1

x1

4

∫ +x

xdx = π

1

x1

4

∫ +1dx

π ln x + x⎡⎣ ⎤⎦1

4= π ln 4 + 4 − ln1−1[ ] = π (ln 4 + 3)

http://clem.mscd.edu/~talmanl/HTML/DetailedVolRev.html

Sphere Torus

x

y

x

y

What would be these Solids of Revolution about the x-axis?

Sphere Torus

x

y

x

y

What would be these Solids of Revolution about the x-axis?

• http://curvebank.calstatela.edu/volrev/volrev.htm