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Today's Calculation Of Integral
2012 Today's Calculation Of Integral
855
Today's calculation of Integral 855
Today's calculation of Integral 855
Source:
November 17, 2012
calculus
integration
function
limit
calculus computations
Problem Statement
Let
f
(
x
)
f(x)
f
(
x
)
be a function which is differentiable twice and
f
′
′
(
x
)
>
0
f''(x)>0
f
′′
(
x
)
>
0
on
[
0
,
1
]
[0,\ 1]
[
0
,
1
]
.For a positive integer
n
n
n
, find
lim
n
→
∞
n
{
∫
0
1
f
(
x
)
d
x
−
1
n
∑
k
=
0
n
−
1
f
(
k
n
)
}
.
\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.
lim
n
→
∞
n
{
∫
0
1
f
(
x
)
d
x
−
n
1
∑
k
=
0
n
−
1
f
(
n
k
)
}
.
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