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Today's Calculation Of Integral
2011 Today's Calculation Of Integral
717
717
Part of
2011 Today's Calculation Of Integral
Problems
(1)
Today's calculation of Integral 717
Source: 1978 Osaka City University entrance exam
6/26/2011
Let
a
n
a_n
a
n
be the area of the part enclosed by the curve
y
=
x
n
(
n
≥
1
)
y=x^n\ (n\geq 1)
y
=
x
n
(
n
≥
1
)
, the line
x
=
1
2
x=\frac 12
x
=
2
1
and the
x
x
x
axis. Prove that :
0
≤
ln
2
−
1
2
−
(
a
1
+
a
2
+
⋯
⋯
+
a
n
)
≤
1
2
n
+
1
0\leq \ln 2-\frac 12-(a_1+a_2+\cdots\cdots+a_n)\leq \frac {1}{2^{n+1}}
0
≤
ln
2
−
2
1
−
(
a
1
+
a
2
+
⋯⋯
+
a
n
)
≤
2
n
+
1
1
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