Use a computer algebra system and the result "The centroid of the region having area $A$ bounded by the simple closed path $C$ is $\bar { x } = \frac { 1 } { 2 A } \int _ { C } x ^ { 2 } d y , \bar { y } = - \frac { 1 } { 2 A } \int _ { C } y ^ { 2 } d x ^ { \prime \prime }$ to find the centroid of the region bounded by the graphs of $y = \sqrt { 196 - x ^ { 2 } }$ and $y = 0$.
A) $\left( 0 , \frac { 56 } { 3 \pi } \right)$
B) $( 28 \pi , 0 )$
C) $( 0,56 \pi )$
D) $\left( 0 , \frac { 14 } { 3 \pi } \right)$
E) $\left( \frac { 56 } { 3 \pi } , 0 \right)$

Question

Use a computer algebra system and the result "The centroid of the region having area $A$ bounded by the simple closed path $C$ is $\bar { x } = \frac { 1 } { 2 A } \int _ { C } x ^ { 2 } d y , \bar { y } = - \frac { 1 } { 2 A } \int _ { C } y ^ { 2 } d x ^ { \prime \prime }$ to find the centroid of the region bounded by the graphs of $y = \sqrt { 196 - x ^ { 2 } }$ and $y = 0$. 
A) $\left( 0 , \frac { 56 } { 3 \pi } \right)$
B) $( 28 \pi , 0 )$
C) $( 0,56 \pi )$
D) $\left( 0 , \frac { 14 } { 3 \pi } \right)$
E) $\left( \frac { 56 } { 3 \pi } , 0 \right)$

Quizplus · Accepted Answer

The region described is a semicircle with radius $r = \sqrt{196} = 14$, centered at the origin and above the x-axis. The area $A$ of this semicircle is $\frac{1}{2} \pi r^2 = \frac{1}{2} \pi (14)^2 = 98\pi$. The centroid of a semicircle is known to lie on the y-axis (hence, $\bar{x} = 0$) and at a distance of $\frac{4r}{3\pi}$ from the base along the y-axis. Substituting $r = 14$ gives $\bar{y} = \frac{4(14)}{3\pi} = \frac{56}{3\pi}$, making the centroid $\left(0, \frac{56}{3\pi}\right)$.

Use a Computer Algebra System and the Result "The Centroid AAA

Use a Computer Algebra System and the Result "The Centroid $A$