Find the length of the indicated portion of the curve.
-$$r ( t ) = \left( e ^ { t } \cos t \right) i + \left( e ^ { t } \sin t \right) j + 7 e ^ { t } k ^ { \prime } , - \ln 2 \leq t \leq 0$$
A) $\frac { \sqrt { 47 } } { 2 }$
B) $\frac { \sqrt { 51 } } { 2 }$
C) $\frac { \sqrt { 47 } } { 4 }$
D) $\frac { \sqrt { 51 } } { 4 }$

Question

Quizplus · Accepted Answer

The length of the curve is calculated using the integral of the magnitude of the derivative of the vector function. The derivative $ r'(t) $ is calculated, and its magnitude is integrated over the interval $[- \ln 2, 0]$. The correct calculation results in $\frac{\sqrt{51}}{2}$.

Find the Length of the Indicated Portion of the Curve r(t)=(etcos⁡t)i+(etsin⁡t)j+7etk′,−ln⁡2≤t≤0r ( t ) = \left( e ^ { t } \cos t \right) i + \left( e ^ { t } \sin t \right) j + 7 e ^ { t } k ^ { \prime } , - \ln 2 \leq t \leq 0r(t)=(etcost)i+(etsint)j+7etk′,−ln2≤t≤0

Find the Length of the Indicated Portion of the Curve $r ( t ) = \left( e ^ { t } \cos t \right) i + \left( e ^ { t } \sin t \right) j + 7 e ^ { t } k ^ { \prime } , - \ln 2 \leq t \leq 0$