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Question 1015a
Definite Integrals: Areas and Volumes
Question
Generate new
(i)
Given that
f
{f}
f
is a continuous function, explain, with the aid of a sketch, why the value of
lim
n
→
∞
1
n
{
f
(
1
n
)
+
f
(
2
n
)
+
⋯
+
f
(
n
n
)
}
\lim_{n \to \infty} \frac{1}{n} \left\{ f \left( \frac{1}{n} \right) + f \left( \frac{2}{n} \right) + \cdots + f \left( \frac{n}{n} \right) \right\}
n
→
∞
lim
n
1
{
f
(
n
1
)
+
f
(
n
2
)
+
⋯
+
f
(
n
n
)
}
is
∫
0
1
f
(
x
)
d
x
.
{\displaystyle \int_0^1 f(x) \; \mathrm{d}x.}
∫
0
1
f
(
x
)
d
x
.
[2]
(ii)
Hence evaluate
lim
n
→
∞
1
n
(
sin
(
1
n
)
+
sin
(
2
n
)
+
⋯
+
sin
(
n
n
)
)
.
\lim_{n\to\infty} \frac{1}{n} \left( \sin \left( \frac{1}{n} \right)+\sin \left( \frac{2}{n} \right)+\cdots+\sin \left( \frac{n}{n} \right) \right).
n
→
∞
lim
n
1
(
sin
(
n
1
)
+
sin
(
n
2
)
+
⋯
+
sin
(
n
n
)
)
.
[3]
Answer
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