Find the x-coordinate of the center of mass of the lamina that occupies the part of the disk $x ^ { 2 } + y ^ { 2 } \leq 1$ in the first quadrant and has density function p(x, y) = xy.
A) $\frac { 15 } { 8 }$

B) $\frac { 1 } { 5 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 8 } { 15 }$
F) $\frac { 1 } { 8 }$

G) $\frac { 5 } { 6 }$

H)1

Question

Find the x-coordinate of the center of mass of the lamina that occupies the part of the disk $x ^ { 2 } + y ^ { 2 } \leq 1$ in the first quadrant and has density function p(x, y) = xy.
A) $\frac { 15 } { 8 }$ 
 
B) $\frac { 1 } { 5 }$ 
C) $\frac { 1 } { 3 }$ 
D) $\frac { 1 } { 2 }$ 
E) $\frac { 8 } { 15 }$
F) $\frac { 1 } { 8 }$

G) $\frac { 5 } { 6 }$

H)1

Quizplus · Accepted Answer

The x-coordinate of the center of mass ($\bar{x}$) for a lamina is given by $\bar{x} = \frac{1}{M} \int \int_R x \rho(x,y) dA$, where $R$ is the region occupied by the lamina, $\rho(x,y)$ is the density function, and $M$ is the total mass of the lamina. For a quarter disk in the first quadrant with $x^2 + y^2 \leq 1$ and $\rho(x,y) = xy$, the calculation involves integrating over the quarter circle. The correct answer, $\frac{8}{15}$, results from setting up and evaluating the double integral for the x-coordinate of the center of mass correctly, taking into account the given density function and the limits of integration that correspond to the quarter disk.

Find the X-Coordinate of the Center of Mass of the Lamina