Find the center of mass of a thin plate covering the given region with the given density function.
-The region bounded by the parabola $y = 64 - x ^ { 2 }$ and the $x$-axis, with density $\delta ( x ) = 8 x ^ { 2 }$
A) $\bar { x } = \frac { 128 } { 5 } , \bar { y } = 0$
B) $\bar { x } = 0 , \bar { y } = \frac { 128 } { 7 }$
C) $\bar { x } = 0 , \bar { y } = \frac { 128 } { 5 }$
D) $\bar { x } = 0 , \bar { y } = \frac { 256 } { 7 }$

Question

Quizplus · Accepted Answer

The center of mass $\bar{x}$ is 0 due to symmetry of the region and density function about the y-axis. The calculation for $\bar{y}$ involves integrating the moment about the x-axis and dividing by the total mass, leading to $\bar{y} = \frac{128}{7}$.

Find the Center of Mass of a Thin Plate Covering y=64−x2y = 64 - x ^ { 2 }y=64−x2

Find the Center of Mass of a Thin Plate Covering $y = 64 - x ^ { 2 }$