Find the center of mass of a thin plate of constant density covering the given region.
-The region between the curve $y = \frac { 4 } { \sqrt { x } }$ and the $x$-axis from $x = 1$ to $x = 16$
A) $\bar { x } = 7 , \bar { y } = \frac { 2 } { 3 } \ln 16$
B) $\bar { x } = 168 , \bar { y } = 16 \ln 16$
C) $\bar { x } = \frac { 2 } { 3 } \ln 16 , \bar { y } = 7$
D) $\bar { x } = \frac { 1 } { 7 } , \bar { y } = \frac { 3 } { 2 } \ln 16$

Question

Quizplus · Accepted Answer

The center of mass $(\bar{x}, \bar{y})$ for a region can be found using the formulas $\bar{x} = \frac{1}{A}\int_{a}^{b} x \cdot f(x) dx$ and $\bar{y} = \frac{1}{2A}\int_{a}^{b} [f(x)]^2 dx$, where $A$ is the area of the region. For the given region, the area $A$ is found by integrating $y = \frac{4}{\sqrt{x}}$ from $x = 1$ to $x = 16$, which results in $A = 16$. Calculating $\bar{x}$ and $\bar{y}$ with these formulas gives $\bar{x} = 7$ and $\bar{y} = \frac{2}{3}\ln 16$, matching option A.

Find the Center of Mass of a Thin Plate of Constant