Find the center of mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $y = 1 - x ^ { 2 }$ and the $x$-axis.
$$\rho ( x , y ) = 4 y$$
A) $( 8,4 )$
B) $( 4,46.44 )$
C) $( 0,0.57 )$
D) $( 12,0.57 )$
E) None of these

Question

Quizplus · Accepted Answer

The center of mass $(\bar{x}, \bar{y})$ is calculated using the formulas $\bar{x} = \frac{1}{M} \int \int_D x \rho(x, y) \, dA$ and $\bar{y} = \frac{1}{M} \int \int_D y \rho(x, y) \, dA$, where $M$ is the total mass. For this problem, the calculations yield $(\bar{x}, \bar{y}) = (0, 0.57)$.

Find the Center of Mass of the Lamina That Occupies DDD

Find the Center of Mass of the Lamina That Occupies $D$