Find the x-coordinate of the centroid of the region bounded by the graphs $y = \sqrt [ 3 ] { x } , \quad x = 8$ and the x-axis.
A)8
B) $\frac { 16 } { 7 }$
C) $\frac { 7 } { 16 }$
D) $\frac { 224 } { 9 }$
E)4

F) $\frac { 32 } { 7 }$
G) $\frac { 7 } { 32 }$
H) $\frac { 2 } { 3 }$

Question

Find the x-coordinate of the centroid of the region bounded by the graphs $y = \sqrt [ 3 ] { x } , \quad x = 8$ and the x-axis.
A)8
B) $\frac { 16 } { 7 }$ 
C) $\frac { 7 } { 16 }$ 
D) $\frac { 224 } { 9 }$ 
E)4

F) $\frac { 32 } { 7 }$ 
G) $\frac { 7 } { 32 }$ 
H) $\frac { 2 } { 3 }$

Quizplus · Accepted Answer

The x-coordinate of the centroid is given by $\frac{1}{A} \int_{a}^{b} x f(x) \, dx$, where $A$ is the area under the curve. For $y = \sqrt[3]{x}$ from $x = 0$ to $x = 8$, the area $A = \int_{0}^{8} \sqrt[3]{x} \, dx = \frac{24}{5}$. The x-coordinate of the centroid is $\frac{1}{A} \int_{0}^{8} x \sqrt[3]{x} \, dx = \frac{32}{7}$.

Find the X-Coordinate of the Centroid of the Region Bounded y=x3,x=8y = \sqrt [ 3 ] { x } , \quad x = 8y=3x​,x=8

Find the X-Coordinate of the Centroid of the Region Bounded $y = \sqrt [ 3 ] { x } , \quad x = 8$