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Lecture 20:Volumes
Part 1.Volume: The Disk Method
1．The Disk Method
When a region in the plane is revolved about a line, the
resulting solid is a solid of revolution, and the line is called the
axis of revolution. The simplest such solid is a right circular
cylinder or disk, which is formed by revolving a rectangle
about an axis adjacent to one side of the rectangle, as shown in
Figure . The volume of such a disk is
Volume of disk = (area of disk)(width of disk) = πR2w,
where R is the radius of the disk and w is he width.
Consider a solid of revolution formed by revolving the plane region in Figure about the indicated
axis.To determine the volume of this solid, consider a representative rectangle in the plane
region. When this rectangle is revolved about the axis of revolution, it generates a representative
disk whose volume is V =  R2 x .Approximating the volume of the solid by n such disks of
n
n
width x and radius R( xi ) produces Volume of solid   [ R( xi )] x =  [ R( xi )] x .
i =1
2
2
i =1
So, we can define the volume of the solid as Volume of solid =
n
b
i =1
a
lim  [ R( xi )]2 x =   [ R( x)]2 dx .
x →0
2．Definition of Volume
(1)Let S be a solid that lies between x = a and x = b . If the cross-sectional area of S in the plane
Px , through x and perpendicular to the x-axis, is A( x) , where A is a continuous function, then
the volume of S is
V =  A( x) dx
b
a
(2)Let S be a solid that lies between y = c and y = d . If the cross-sectional area of S in the
plane Py , through y and perpendicular to the yx-axis, is A( y ) , where A is a continuous function,
then the volume of S is
V =  A( y ) dy
d
c
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【例 1】
1 ． Find the volume of the solid formed by revolving the region bounded by the graph of
f ( x) = sin x and the x-axis ( 0≤x≤ ) about the x-axis, as shown in Figure.
2．Find the volume obtained when the shaded region is rotated through 360°about the x-axis.
3．Find the volume obtained when the shaded region is rotated through 360°about the y-axis.
2
4．The region R is bounded by the curve with equation y = sin 2 x ,the x-axis and the lines x = 0
and x =

.
2
(a) Find the area of R.
(b) Find the volumn of the solid formed when the region R is rotated through 2 radians
about the x-axis.
【例 2】
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1．Show that the volume of a sphere of radius r is V =  r 3 .
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3
2．Find the volume of the solid obtained by rotating the region bounded by y = x 3 , y = 8 , and
x = 0 about the y-axis.
3．Find the volume of the solid formed by revolving the region bounded by the graphs of
f ( x ) = 2 − x 2 and g ( x) = 1 about the line y = 1 .
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4．Find the volumes of the solids (see figures) generated if the upper half of the ellipse
x2 y2
+
= 1 is revolved about the x-axis to form a prolate spheroid.
a 2 b2
Part 2.Volume: The Washer Method
In general, we calculate the volume of a solid of revolution by using the basic defining formula
V =  A( x) dx or V =  A( y ) dy
b
d
a
c
and we find the cross-sectional area A( x) or A( y ) in one of the following ways:
• If the cross-section is a disk, we find the radius of the disk (in terms of x or y) and use
A =  (radius)2
• If the cross-section is a washer, we find the inner radius rin and outer radius rout from a sketch and
compute the area of the washer by subtracting the area of the inner disk from the area of the outer
disk:
A =  (outer radius)2 −  (inner radius)2
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【例 3】
1．Find the volume of the solid obtained when the shaded region is rotated through 360°about
the x-axis.
2．The region R enclosed by the curves y = x and y = x 2 is rotated about the x-axis. Find the
volume of the resulting solid.
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3．Find the volume of the solid formed by revolving the region bounded by the graphs of y = x
and y = x 2 about the x-axis
2
.
x
The line y = 7 intersects the curve at the points P and Q.
4．The diagram shows part of the curve y = 3x +
(a) Find the coordinates of P and Q.
(b) Find the volume obtained when the shaded region is rotated through 360°about the x-axis.
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【例 4】
1．Figure shows a solid with a circular base of radius 1. Parallel cross-sections perpendicular to
the base are equilateral triangles. Find the volume of the solid.
2．Find the volume of a pyramid whose base is a square with side L and whose height is h.
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【例 5】
1．Find the volume of the solid formed by revolving the region bounded by the graphs of
y = x 2 + 1 , y = 0 , x = 0 , and x = 1 about the y-axis.
1
2．The diagram shows part of the curve y = (1 + 4 x) 2 and a point P(6, 5) lying on the curve. The
line PQ intersects the x-axis at Q(8, 0).
(a) Show that PQ is a normal to the curve.
(b) Find, showing all necessary working, the exact volume of revolution obtained when the
shaded region is rotated through 360°about the x-axis.
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【例 6】
1．A wedge is cut out of a circular cylinder of radius 4 by two planes. One plane is perpendicular
to the axis of the cylinder. The other intersects the first at an angle of 30°along a diameter of
the cylinder. Find the volume of the wedge.
2．(a) Set up an integral for the volume of a solid torus (the donut-shaped solid shown in the
figure) with radii r and R.
(b) By interpreting the integral as an area, find the volume of the torus.
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Exercises
1．Find the volume obtained when the shaded region is rotated through 360°about the x-axis.
2．Find the volume obtained when the shaded region is rotated through 360°about the y-axis.
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3．The diagram shows part of the curve y = x3 + 4 x 2 + 3x + 2 . Find the volume obtained when
the shaded region is rotated through 360°about the x-axis.
2
. The shaded area is rotated through 360°
2x +1
about the x-axis between x = 0 and x = p . Show that as p →  , the volume approaches the
value 2π.
4．The diagram shows part of the curve y =
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5．The diagram shows part of the curve y = 25 − x 2 . The point P(4, 3) lies on the curve.
(a) Find the volume obtained when the shaded region is rotated through 360°about the y-axis.
(b) Find the volume obtained when the shaded region is rotated through 360°about the x-axis.
6．The diagram shows the curve y = 4 − x and the line x + 2 y = 4 that intersect at the points
(4, 0) and (0, 2).
(a) Find the volume obtained when the shaded region is rotated through 360°about the x-axis.
(b) Find the volume obtained when the shaded region is rotated through 360°about the y-axis.
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7．A mathematical model for the inside of a bowl is obtained by rotating the curve x 2 + y 2 = 100
through 360°about the y-axis between y = −8 and y = 0 . Each unit of x and y represents 1cm.
(a) Find the volume of the bowl. The bowl is filled with water to a depth of 3 cm.
(b) Find the volume of water in the bowl.
8．Find the volume of the solid obtained by rotating the region bounded by the given curves
about the specified line. Sketch the region, the solid, and a typical disk or washer.
(a) y = e x , y = 0 , x = −1 , x = 1 ; about the x-axis.
(b) y = sin x , y = cos x , 0≤x≤

4
; about y = −1 .
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9．Find the volume of the described solid S.
(a)A right circular cone with height h and base radius r.
(b)A frustum of a right circular cone with height h, lower base radius R, and top radius r.
(c)A cap of a sphere with radius r and height h.
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10．Find the volume of the described solid S.
The base of S is the region enclosed by y = 2 − x 2 and the x-axis. Cross-sections
perpendicular to the y-axis are quarter-circles.
11．Find the volume of the described solid S.
The base of S is an elliptical region with boundary curve 9 x 2 + 4 y 2 = 36 . Cross-sections
perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.
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