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.

---

Definitions:

Supervisors

Individuals in charge of overseeing and directing the work of others, typically in a workplace setting.

Production workers

Individuals involved in the manufacturing process of goods, responsible for operating machinery, assembling products, and performing quality checks.

Currency Exchange Rates

The monetary value of one currency in the form of another during conversion.

Equivalent Amount

A sum of money that has the same value as another, especially in different currencies or at different times, adjusted for factors like inflation.