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Mathematics
Problem
For the spiral
r
=
a
θ
r=a\theta
r
=
a
θ
with
a
>
0
a>0
a
>
0
, which expression gives the exact length of the curve from
θ
=
0
\theta=0
θ
=
0
to
θ
=
2
π
\theta=2\pi
θ
=
2
π
? a)
a
[
2
π
4
π
2
+
1
+
1
2
ln
(
2
π
+
4
π
2
+
1
)
]
a\left[2\pi\sqrt{4\pi^{2}+1}+\dfrac12\ln\!\left(2\pi+\sqrt{4\pi^{2}+1}\right)\right]
a
[
2
π
4
π
2
+
1
+
2
1
ln
(
2
π
+
4
π
2
+
1
)
]
b)
a
[
π
4
π
2
+
1
+
1
2
ln
(
2
π
+
4
π
2
+
1
)
]
a\left[\pi\sqrt{4\pi^{2}+1}+\dfrac12\ln\!\left(2\pi+\sqrt{4\pi^{2}+1}\right)\right]
a
[
π
4
π
2
+
1
+
2
1
ln
(
2
π
+
4
π
2
+
1
)
]
c)
a
[
π
4
π
2
+
1
+
ln
(
2
π
+
4
π
2
+
1
)
]
a\left[\pi\sqrt{4\pi^{2}+1}+\ln\!\left(2\pi+\sqrt{4\pi^{2}+1}\right)\right]
a
[
π
4
π
2
+
1
+
ln
(
2
π
+
4
π
2
+
1
)
]
d)
a
[
π
4
π
2
+
1
]
a\left[\pi\sqrt{4\pi^{2}+1}\right]
a
[
π
4
π
2
+
1
]
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