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Integrals of standard functions

Reversing the chain rule

Integration by substitution

Volumes of revolution

Examination-style question




Consider the area bounded by the curve y = f(x), the x-axis and x = a and x = b.

If this area is rotated 360° about the x-axis a three-dimensional shape called a solid of revolution is formed.

The volume of this solid is called its volume of revolution.



We can calculate the volume of revolution by dividing the volume of revolution into thin slices of width δx.

The volume of each slice is approximately cylindrical, of radius y and height δx, and is therefore approximately equal to

πy2δx



The total volume of the solid is given by the sum of the volume of the slices. =

2

=

x b

x a

V y x

The smaller δx is, the closer this approximate area is to the actual area.

We can find the actual area by considering the limit of this sum as δx tends to 0. =

2

0=

= limx b

xx a

V y x

This limit is represented by the following integral:

2=b

aV y dx



So in general, the volume of revolution V of the solid generated by rotating the curve y = f(x) between x = a and x = b about the x-axis is:

Similarly, the volume of revolution V of the solid generated by rotating the curve x = f(y) between y = a and y = b about the y-axis is:

Volumes of revolution are usually given as multiples of π.

2=b

aV y dx

2=b

aV x dy



Find the volume of the solid formed by rotating the area between the curve y = x(2 – x), the x-axis, x = 0, and x = 2 360° about the x-axis.

2 2

0=V y dx

2 2 2

0= (2 )x x dx

2 2 3 4

0= (4 4 )x x x dx

23 4 5

0

4 4=

3 4 5

x x x

32 32= 16 +

3 5

16cubic u s

15= nit



2 2

1=V x dy

22

1

1= dy

y

2 2

1= y dy

2

1

1=

x

1= +1

2

cubic u= nits

2

Find the volume of the solid formed by rotating the area between the curve y = , the y-axis, y = 1, and y = 2 360° about the y-axis.

1x

Rearranging y = gives x = .

1x

1y


Rotating regions between curves

If two graphs, and intersect at and

and in the interval then the volume of the solid of revolution formed by rotating the region between the graphs about the x-axis is

)()( xgxf )(xfy )(xgy ax

bx ,bxa

dxxgdxxfb

a

b

a 22 ))(())((

Or b

adxxgxf 22 ))(())((


The graphs intersect when , that is, when or when

Example

Find the volume obtained when the minor segment between the circle and the chord is rotated about the y-axis.

2522 yx 4x

1625 2 y,92 y .3y

The required volume V is then given by

dydyyV

3

3

23

3

2 4)25(

dyy )9(3

3

2 3

3

3

3

19

yy


Example cont’d

)))27(3

1)27(()27

3

127((

)18()18(

36

The required volume is 36

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