SPHERES
Show that the volume of a sphere
of radius r is
If we place the sphere so that its
center is at the origin, then the
plane
P
x
intersects the sphere in a
circle whose radius, from the
Pythagorean Theorem, is:
So, the crosssectional area is:
Using the definition of volume with
a
=
r
and
b
=
r
, we have:
2
2
y
r
x
=

3
4
3
.
V
r
π
=
Example 1 (6.2)
2
2
2
( )
(
)
A x
y
r
x
=
=

( )
2
2
2
2
0
3
3
2
3
0
3
4
3
( )
2
(
)
2
2
3
3
r
r
r
r
r
r
V
A x dx
r
x
dx
r
x
dx
x
r
r x
r
r


=
=

=

=

=

=
∫
∫
∫
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View Full DocumentFind the volume of the solid obtained by rotating about the xaxis the
region under the curve
from 0 to 1.
Illustrate the definition of volume by sketching a typical approximating
cylinder.
The region is shown in the first figure. If we rotate about the
x
axis, we
get the solid shown in the next figure.
s
When we slice through the point
x
, we get a disk with radius
√
x .
The area of the crosssection is:
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 Fall '08
 Noohi
 Calculus, Pythagorean Theorem, Euclidean geometry, crosssectional area

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