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Mathematics
Problem
A composite curve
C
\mathcal{C}
C
consists of two segments: •
C
1
:
x
=
y
2
C_{1}: x=y^{2}
C
1
:
x
=
y
2
for
0
≤
y
≤
1
0\le y\le 1
0
≤
y
≤
1
•
C
2
C_{2}
C
2
: the horizontal line segment from
(
1
,
1
)
(1,1)
(
1
,
1
)
to
(
3
,
1
)
(3,1)
(
3
,
1
)
What is the exact total length of
C
\mathcal{C}
C
? a)
2
+
5
2+\sqrt{5}
2
+
5
b)
1
2
5
+
2
\displaystyle \frac12\sqrt{5}+2
2
1
5
+
2
c)
1
2
5
+
1
4
ln
(
2
+
5
)
\displaystyle \frac12\sqrt{5}+\frac14\ln\bigl(2+\sqrt{5}\bigr)
2
1
5
+
4
1
ln
(
2
+
5
)
d)
1
2
5
+
1
4
ln
(
2
+
5
)
+
2
\displaystyle \frac12\sqrt{5}+\frac14\ln\bigl(2+\sqrt{5}\bigr)+2
2
1
5
+
4
1
ln
(
2
+
5
)
+
2
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