Transcript 7.3 Volumes of Solids with Known Cross Sections

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( x)dx
a
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
3
-3
dx
As
2
A  (2 y )
2
A  4y
y
2
y-coordinate

So, s = 2y
A 4 9 x
x
2
A  4(9  x )
2
x  y 9
2
2
y  9 x

2
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   A( x)dx
x1
3
-3
dx
x
3
V   4(9  x 2 ) dx
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   4(9  x 2 ) dx
y
3
3
V  4  (9  x 2 ) dx
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
2
y2
ellipse x

1
4 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
a 2 y
2
y
2
  
2
25x 2
A( x )  25 
4
2
 
-5
1 2
A( x )  a
2
1
A( x ) 
2 y
2
A( x )  y 2
2
2.) Set up & evaluate the
integral.

25x 2   200 units 3
2 25  4 dx 3
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)?
1 2
A  r
2
2
4
1  1  1

A     x  2  
2 2 2

x
r is half of the yvalue on the line
y  m x b
2
1  1

A     x  1
2  4

y
2
1
x2
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)?
1  1

A     x  1
2  4

2
2
Volume
2
1  1

V      x  1 dx
2 0 4

4
4
x
(fInt)
V = 2.094