Find the coordinates of the centroid of the curve.
-Find the coordinates of the centroid of the curve $x = e ^ { 5 t } \cos 5 t , y = e ^ { 5 t } \sin 5 t , 0 \leq t \leq \frac { \pi } { 10 }$.
A) $( \bar { x } , \bar { y } ) = \left( \frac { e ^ { \pi + 2 } } { 5 \left( e ^ { \pi / 2 } - 1 \right) } , \frac { 2 e ^ { \pi } - 1 } { 5 \left( e ^ { \pi / 2 } - 1 \right) } \right)$
B) $( \bar { x } , \bar { y } ) = \left( \frac { e ^ { \pi } - 2 } { 5 \left( e ^ { \pi } - 1 \right) } , \frac { 2 e ^ { \pi } + 1 } { 5 \left( e ^ { \pi } - 1 \right) } \right)$
C) $( \bar { x } , \bar { y } ) = \left( \frac { 2 e ^ { \pi } - 1 } { e ^ { \pi / 2 } - 1 } , \frac { e ^ { \pi + 1 } } { e ^ { \pi / 2 } - 1 } \right)$
D) $( \bar { x } , \bar { y } ) = \left\{ \frac { e ^ { \pi } - 2 } { 5 \left( e ^ { \pi / 2 } - 1 \right) } , \frac { 2 e ^ { \pi } + 1 } { 5 \left( e ^ { \pi / 2 } - 1 \right) } \right)$

Question

Quizplus · Accepted Answer

The centroid $(\bar{x}, \bar{y})$ of a curve parameterized by $x(t)$ and $y(t)$ over an interval $a \leq t \leq b$ can be found using the formulas for $\bar{x} = \frac{1}{b-a} \int_{a}^{b} x(t) dt$ and $\bar{y} = \frac{1}{b-a} \int_{a}^{b} y(t) dt$. The given functions and interval lead to the correct calculation as shown in option D, considering the specific integration and normalization over the interval $0 \leq t \leq \frac{\pi}{10}$.

Find the Coordinates of the Centroid of the Curve x=e5tcos⁡5t,y=e5tsin⁡5t,0≤t≤π10x = e ^ { 5 t } \cos 5 t , y = e ^ { 5 t } \sin 5 t , 0 \leq t \leq \frac { \pi } { 10 }x=e5tcos5t,y=e5tsin5t,0≤t≤10π​

Find the Coordinates of the Centroid of the Curve $x = e ^ { 5 t } \cos 5 t , y = e ^ { 5 t } \sin 5 t , 0 \leq t \leq \frac { \pi } { 10 }$