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7.3
VOLUMES WITH KNOWN CROSS
SECTIONS
VOLUMES WITH KNOWN CROSS SECTIONS

A solid has as its base the circle x2 + y2 = 9, and
all cross sections parallel to the y-axis are
squares. Find the volume of the solid.
SOLIDS WITH KNOWN CROSS SECTIONS

If A(x) is the area of a cross section of a solid and A(x)
is continuous on [a, b], then the volume of the solid
from x = a to x = b is
b
V 

a
A ( x ) dx
VOLUMES WITH KNOWN CROSS SECTIONS

A solid has as its base the circle x2 + y2 = 9, and
all cross sections parallel to the y-axis are
squares. Find the volume of the solid.
Area of cross section (square)?
3
-3
y
As
2
x  y 9
2
A  (2 y )
3
-3
dx
A  4y
2
2
y-coordinate
A 4
x

So, s = 2y
y
9x
2
A  4 (9  x )
2

2
2
9 x
2
VOLUMES WITH KNOWN CROSS SECTIONS

A solid has as its base the circle x2 + y2 = 9, and
all cross sections parallel to the y-axis are
squares. Find the volume of the solid.
Area of cross section (square)?
3
-3
y
A  4 (9  x )
2
Volume of solid:
x2
V 
x1
3
-3
dx
 A ( x ) dx
x
3
V 
 4 ( 9  x ) dx
2
3
VOLUMES WITH KNOWN CROSS SECTIONS

A solid has as its base the circle x2 + y2 = 9, and
all cross sections parallel to the y-axis are
squares. Find the volume of the solid.
Volume of solid:
3
3
-3
V 
y
 4 ( 9  x ) dx
2
3
3
V  4  ( 9  x ) dx
2
3
3
1 3

V  4 9 x  x 
3  3

3
-3
dx
x
V  4 18   18 
V  4 36 
V  144
KNOWN CROSS SECTIONS
 Ex:
The base of a solid is the region enclosed by the
ellipse x 2 y 2

4
1
25
The cross sections are perpendicular to the x-axis and
are isosceles right triangles whose hypotenuses are on
the ellipse. Find the volume of the solid.
5
-2
a
a
2
-5
5
1.) Find the area of the cross
section A(x).
-2
a
a
a  a  (2 y )
2
2
2a  4 y
2
y
2
a
-5
A( x) 
1
a
2
1
2
A( x)  y
 2 y 
2
2
A ( x )  25 
2
 2 y
2.) Set up & evaluate the
integral.
2
A( x) 
2
25 x
4
2
2

25 x
  25  4
 2
2
200

units
 dx 

3

3
EXAMPLE

The base of a solid is the region enclosed by the
triangle whose vertices are (0, 0), (4, 0), and (0,
2). The cross sections are semicircles
perpendicular to the x-axis. Find the volume of
the solid. y
Area of cross section (semicircle)?
2
A
1
r
r is half of the yvalue on the line
2
2
 1  1
A    
x
2 2 2
1
4
x

2  

 1

A     x  1
2  4

1
y  mx  b
2
y
1
2
2
x2
EXAMPLE

The base of a solid is the region enclosed by the
triangle whose vertices are (0, 0), (4, 0), and (0,
2). The cross sections are semicircles
perpendicular to the x-axis. Find the volume of
the solid. y
Area of cross section (semicircle)?
2
A
1
2



x  1
4


1

1
2
Volume
V 
4
1
2
x
4
 
0

(fInt)
V = 2.094
2

x  1  dx
4

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