Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $x = \frac { y ^ { 2 } } { 4 }$ and $x = 4 .$ Then $\bar { y }$ =

A)0
B) $\frac { 1 } { 5 }$
C)1
D) $\frac { 3 } { 4 }$
E) $\frac { 45 } { 28 }$

Question

Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $x = \frac { y ^ { 2 } } { 4 }$ and $x = 4 .$ Then $\bar { y }$ =

A)0
B) $\frac { 1 } { 5 }$
C)1
D) $\frac { 3 } { 4 }$
E) $\frac { 45 } { 28 }$

Quizplus · Accepted Answer

The centroid $\bar{y}$ of a region is given by the formula $\bar{y} = \frac{1}{A}\int y \, dA$, where $A$ is the area of the region. For symmetric regions about the y-axis, like the one enclosed by $x = \frac{y^2}{4}$ and $x = 4$, the y-coordinate of the centroid ($\bar{y}$) is 0 because the region's upper half is a mirror image of its lower half, balancing out the positive and negative contributions to $\bar{y}$.

Let yˉ\bar { y }yˉ​ Denote the Y-Coordinate of the Centroid of the Region Enclosed

Let $\bar { y }$ Denote the Y-Coordinate of the Centroid of the Region Enclosed