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Mathematics
Problem
Consider the curve defined by
x
=
t
2
x=t^{2}
x
=
t
2
and
y
=
t
3
3
y=\dfrac{t^{3}}{3}
y
=
3
t
3
for
0
≤
t
≤
2
0\le t \le 2
0
≤
t
≤
2
. Which of the following definite integrals gives the arc length of the curve on this interval? a)
∫
0
2
1
+
t
2
4
d
t
\displaystyle \int_{0}^{2} \sqrt{1+\dfrac{t^{2}}{4}}\,dt
∫
0
2
1
+
4
t
2
d
t
b)
∫
0
2
4
t
2
+
t
4
d
t
\displaystyle \int_{0}^{2} \sqrt{4t^{2}+t^{4}}\,dt
∫
0
2
4
t
2
+
t
4
d
t
c)
∫
0
2
(
4
t
2
+
t
4
)
d
t
\displaystyle \int_{0}^{2} (4t^{2}+t^{4})\,dt
∫
0
2
(
4
t
2
+
t
4
)
d
t
d)
∫
0
2
(
2
t
)
+
(
t
2
)
d
t
\displaystyle \int_{0}^{2} \sqrt{(2t)+(t^{2})}\,dt
∫
0
2
(
2
t
)
+
(
t
2
)
d
t
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