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Putnam
Putnam 1938
A1
A1
Part of
Putnam 1938
Problems
(1)
Putnam 1938 A1
Source:
8/20/2021
A solid in Euclidean
3
3
3
-space extends from
z
=
−
h
2
z = \frac{-h}{2}
z
=
2
−
h
to
z
=
+
h
2
z = \frac{+h}{2}
z
=
2
+
h
and the area of the section
z
=
k
z = k
z
=
k
is a polynomial in
k
k
k
of degree at most
3
3
3
. Show that the volume of the solid is
h
(
B
+
4
M
+
T
)
6
,
\frac{h(B + 4M + T)}{6},
6
h
(
B
+
4
M
+
T
)
,
where
B
B
B
is the area of the bottom
(
z
=
−
h
2
)
(z = \frac{-h}{2})
(
z
=
2
−
h
)
,
M
M
M
is the area of the middle section
(
z
=
0
)
,
(z = 0),
(
z
=
0
)
,
and
T
T
T
is the area of the top
(
z
=
h
2
)
(z = \frac{h}{2})
(
z
=
2
h
)
. Derive the formulae for the volumes of a cone and a sphere.
Putnam