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Mathematics
Mathematics
Problem
Two students are finding the length of the curve
x
(
t
)
=
t
x(t)=t
x
(
t
)
=
t
and
y
(
t
)
=
t
2
y(t)=t^{2}
y
(
t
)
=
t
2
on
0
≤
t
≤
1
0\le t\le1
0
≤
t
≤
1
. Student A:
∫
0
1
1
+
(
2
t
)
2
d
t
\displaystyle \int_{0}^{1}\sqrt{1+(2t)^{2}}\,dt
∫
0
1
1
+
(
2
t
)
2
d
t
Student B:
∫
0
1
t
2
+
(
t
2
)
2
d
t
\displaystyle \int_{0}^{1}\sqrt{t^{2}+(t^{2})^{2}}\,dt
∫
0
1
t
2
+
(
t
2
)
2
d
t
Identify which student's integral correctly represents the arc length and briefly state why.
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