In an Exam, Students Are Asked to Find the Arc

question 53

Essay

In an exam, students are asked to find the arc length of the curve C parameterized by $\vec { r } ( t ) = \left( \frac { t ^ { 2 } } { 2 } - \frac { t ^ { 4 } } { 4 } \right) \vec { i } + \frac { 2 t ^ { 3 } } { 3 } \vec { j } + \vec { k }$ , for $- 1 \leq t \leq 1$ .
One student wrote the following
" $\| \vec { v } ( t ) \| = \sqrt { \left( t - t ^ { 3 } \right) ^ { 2 } + \left( 2 t ^ { 2 } \right) ^ { 2 } } = \sqrt { t ^ { 2 } \left( 1 + 2 t + t ^ { 4 } \right) } = \sqrt { t ^ { 2 } \left( 1 + t ^ { 2 } \right) ^ { 2 } } = t \left( 1 + t ^ { 2 } \right)$ Thus the arc length is $\int _ { - 1 } ^ { 1 } t \left( 1 + t ^ { 2 } \right) d t = 0$ "
This answer cannot be true.
(a)Which part of the student's calculation was wrong?
(b)Find the correct answer.