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Joint Entrance Examination
Mathematics
Indefinite Integrals
Question 88
Question 88 of 416
Easy
MCQ
lim
n
→
∞
3
n
∑
r
=
0
n
−
1
(
2
+
r
n
)
2
\displaystyle\lim_{n\to\infty}\frac3n\sum_{r=0}^{n-1}\left(2+\frac{r}{n}\right)^2
n
→
∞
lim
n
3
r
=
0
∑
n
−
1
(
2
+
n
r
)
2
is equal to:
A
12
B
19
/
3
19/3
19/3
C
0
D
19
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Official solution
This is a Riemann sum:
3
∫
0
1
(
2
+
x
)
2
d
x
=
[
(
2
+
x
)
3
]
0
1
=
27
−
8
=
19
3\displaystyle\int_0^1(2+x)^2\,dx = \left[(2+x)^3\right]_0^1 = 27-8 = 19
3
∫
0
1
(
2
+
x
)
2
d
x
=
[
(
2
+
x
)
3
]
0
1
=
27
−
8
=
19
.
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