Let $\bar { x }$ denote the x-coordinate of the centroid of the region enclosed by and $y ^ { 2 } = 4 x , y = 0,$ $x = 1 \text {, }$ and x=4 Then $\bar { x }$ =

A) $\frac { 1,125 } { 8 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 93 } { 35 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 125 } { 24 }$

Question

Let $\bar { x }$ denote the x-coordinate of the centroid of the region enclosed by and $y ^ { 2 } = 4 x , y = 0,$ $x = 1 \text {, }$ and x=4 Then $\bar { x }$ =

A) $\frac { 1,125 } { 8 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 93 } { 35 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 125 } { 24 }$

Quizplus · Accepted Answer

The x-coordinate of the centroid $\bar{x}$ of a region can be found using the formula $\bar{x} = \frac{1}{A}\int_{a}^{b} x dA$, where $A$ is the area of the region, and $dA$ is a differential area element. For the region bounded by $y^2 = 4x$, $y = 0$, $x = 1$, and $x = 4$, we can express $dA$ as $y dx$ for the parabolic segment. The area $A$ of the region can be found by integrating the function $y = 2\sqrt{x}$ from $x = 1$ to $x = 4$, and then $\bar{x}$ is found by integrating $x \cdot y$ over the same interval and dividing by the area. After performing these integrations and simplifications, the correct answer is $\frac{93}{35}$.

Let xˉ\bar { x }xˉ Denote the X-Coordinate of the Centroid of the Region Enclosed

Let $\bar { x }$ Denote the X-Coordinate of the Centroid of the Region Enclosed