MathDB
Problems
Contests
National and Regional Contests
Japan Contests
Today's Calculation Of Integral
2011 Today's Calculation Of Integral
768
Today's calculation of Integral 768
Today's calculation of Integral 768
Source:
November 17, 2011
calculus
integration
limit
geometry
3D geometry
sphere
symmetry
Problem Statement
Let
r
r
r
be a real such that
0
<
r
≤
1
0<r\leq 1
0
<
r
≤
1
. Denote by
V
(
r
)
V(r)
V
(
r
)
the volume of the solid formed by all points of
(
x
,
y
,
z
)
(x,\ y,\ z)
(
x
,
y
,
z
)
satisfying
x
2
+
y
2
+
z
2
≤
1
,
x
2
+
y
2
≤
r
2
x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2
x
2
+
y
2
+
z
2
≤
1
,
x
2
+
y
2
≤
r
2
in
x
y
z
xyz
x
yz
-space.(1) Find
V
(
r
)
V(r)
V
(
r
)
.(2) Find
lim
r
→
1
−
0
V
(
1
)
−
V
(
r
)
(
1
−
r
)
3
2
.
\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.
lim
r
→
1
−
0
(
1
−
r
)
2
3
V
(
1
)
−
V
(
r
)
.
(3) Find
lim
r
→
+
0
V
(
r
)
r
2
.
\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.
lim
r
→
+
0
r
2
V
(
r
)
.
Back to Problems
View on AoPS