1

Day 6 – Solids of revolutions

If a curve with fixed boundaries is rotated around the x-axis, a 3-dimensional solid is formed.

For example:

Take the function xy , 10 x .

The volume is the product of the area times the height.

hAV (area of a disk)x(height)

Each circle has radius y as x goes from zero to 1.

The area of one circle (disk) is 2y . The width of each circle is the change in x from one circle to another.

Therefore, we can sum the areas of the circles using the formula:

n

k

xxAV1

)( , where n is the number of circles and x is the width of each circle.

If we allow the number of intervals to increase, the width of the intervals to approach zero, then the

limit of this sum as 0x is:

For each disk, the area is 2y , but using the function, we know that xy , therefore we can write:

30

3

1

3

1

0

31

0

2

xdxxV .

Notice that we got the volume of the shape to be hr 2

3

1 , which happens to be the equation for volume of a

cone.

If the curve is rotated around the x-

axis, a cone is formed. We can find

the volume of this shape by slicing it

into cylindrical sections (circles) and

summing their areas.

xy

b

a

b

a

dxxfdxxAV2

)()(

2

When a plane region enclosed by the curve y f x and the lines x a and x b is revolved

about the x-axis, the volume, V 3units , of the solid formed is given by :

2

( )

x b

x a

V f x dx

or 2

b

a

V y dx

When a plane region enclosed by the curve y f x and the lines x a and x b is revolved

about the y-axis, the volume, V 3units , of the solid formed is given by :

2

1( )

y f

y e

V f y dy

or 2

f

e

V x dy

A solid of revolution is one generated by rotating a plane region about a line that lies in the same plane as

the region. For example, if the line 3y , 42 x , is rotated about the x-axis, what solid is formed?

3y

2 4

1829)24(9993)(4

2

4

2

4

2

4

2

4

2

22 xdxdxdxydxxAV

Ex. Write the integral for the volume of the plane region defined by the function ry from x=a to x=b

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Ex. Find the volume of the solid obtained by rotating about the x-axis the region under the curve xy

from 0 to 1.

Ex. The curve 1,1 5y x x is rotated about the x-axis to form a solid of revolution. Sketch this

solid and find its volume.

Ex. The curve 1,1 5y x x is rotated about the y-axis. Sketch this second solid and find its

volume.

4

Ex. Find the volume of the solid of revolution formed by rotating the part of the curve xy e between

x=1 and x=5 about the y-axis.

Ex. Find the volume of the solid formed by revolving the region enclosed by the curve 225f x x

and the line 3g x about the x-axis.