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Mathematics
Problem
For constants
a
>
0
a>0
a
>
0
and
k
>
0
k>0
k
>
0
, the curve
y
=
k
x
2
y=kx^{2}
y
=
k
x
2
is considered on
[
−
a
,
a
]
[-a,a]
[
−
a
,
a
]
. Which of the following gives the exact length of this curve? a)
a
2
1
+
4
k
2
a
2
+
1
4
k
arcsinh
(
2
k
a
)
\displaystyle\frac{a}{2}\sqrt{1+4k^{2}a^{2}}+\frac{1}{4k}\operatorname{arcsinh}(2ka)
2
a
1
+
4
k
2
a
2
+
4
k
1
arcsinh
(
2
k
a
)
b)
a
1
+
4
k
2
a
2
+
1
2
k
arcsinh
(
2
k
a
)
\displaystyle a\sqrt{1+4k^{2}a^{2}}+\frac{1}{2k}\operatorname{arcsinh}(2ka)
a
1
+
4
k
2
a
2
+
2
k
1
arcsinh
(
2
k
a
)
c)
a
k
1
+
4
k
2
a
2
+
1
2
k
arcsinh
(
2
k
a
)
\displaystyle\frac{a}{k}\sqrt{1+4k^{2}a^{2}}+\frac{1}{2k}\operatorname{arcsinh}(2ka)
k
a
1
+
4
k
2
a
2
+
2
k
1
arcsinh
(
2
k
a
)
d)
a
1
+
k
2
a
2
+
1
k
arcsinh
(
k
a
)
\displaystyle a\sqrt{1+k^{2}a^{2}}+\frac{1}{k}\operatorname{arcsinh}(ka)
a
1
+
k
2
a
2
+
k
1
arcsinh
(
k
a
)
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