Volume of Revolution
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Volume of Revolution
We’ll first look at the area between the linesy = x , . . . • x = 1, . . . • and the x-axis.
• Can you see what shape you will get if you rotate the area through about the x-axis?
x = 1, . . . hrV 2
31
We’ll first look at the area between the linesy = x , . . .
and the x-axis.
The formula for the volume found by rotating any area about the x-axis is
a b
x
)(xfy
dxyVb
a 2
where is the curve forming the upper edge of the area being rotated.
)(xfy
a and b are the x-coordinates at the left- and right-hand edges of the area.
We leave the answers in terms of
r
h 0 1
So, for our cone, using integration, we get
dxV 1
0 2x
1
0
3
3
x
031
31
xy
We must substitute for y using before we integrate. dxyV
b
a 2
x
y
The formula can be proved by splitting the area into narrow strips. . . which are rotated about the x-
axis.Each tiny piece is approximately a cylinder ( think of a penny on its side ).
Each piece, or element, has a volume
hr 2 2y dxThe formula comes from adding an infinite number of these elements.
dxyVb
a 2
Solution: To find a volume we don’t need a sketch unless we are not sure what limits of integration we need. However, a sketch is often helpful.As these are the first examples I’ll sketch the curves.
)1( xxy e.g. 1(a) The area formed by the curve and the x-axis from x = 0 to x = 1 is rotated through radians about the x-axis. Find the volume of the solid formed.
2
)1( xxy
dxyVb
a 2
area rotate about the x-axis
A common error in finding a volume is to get wrong. So beware!
2y
)1( xxy 222 )1( xxy
)21( 222 xxxy 4322 2 xxxy
(a) rotate the area between.10)1( tofrom axis- the and xxxy
)1( xxy
dxyVb
a 24322 2 xxxy
dxxxxV 1
0432 2
a = 0, b = 1
(a) rotate the area between.10)1( tofrom axis- the and xxxy
dxxxxV 1
0432 2
1
0
543
542
3
xxx
0
51
21
31
301
30
2
Volumes of Revolution
To rotate an area about the y-axis we use the same formula but with x and y swapped.
dxyVb
a 2 dyxVd
c 2
The limits of integration are now values of y giving the top and bottom of the area that is rotated.
Rotation about the y-axis
As we have to substitute for x from the equation of the curve we will have to rearrange the equation.
Volumes of Revolution
xy
dyxVd
c 2
e.g. The area bounded by the curve , the y-axis and the line y = 2 is rotated through about the y-axis. Find the volume of the solid formed.
xy 360
2y
dyyV 2
04
xy 2yx 42 yx
Volumes of Revolution
dyyV 2
04
2
0
5
5
y
0
5
52
532
Exercise
2xy 0xthe y-axis and the line y = 3 is rotated through radians about the y-axis. Find the volume of the solid formed.2
1(a) The area formed by the curve for
xy 1
(b) The area formed by the curve , the y-axis and the lines y = 1 and y = 2 is rotated 2through radians about the y-axis. Find the volume of the solid formed.
2xy
Solutions: 2xy (a) for , the y-axis and the line y =
3. 0x
3
0dyyV
dyxVd
c 2
3
0
2
2
y2
9
Solution:
2
1 21 dyy
V
dyxVd
c 2
2
1
1
y
1
21
(b)x
y 1
, the y-axis and the lines y = 1 and y = 2.
2
yx
xy 11
22 1
yx
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