Let D Be a Region Bounded by a Simple Closed

Multiple Choice

Let D be a region bounded by a simple closed path C in the xy. Then the coordinates of the centroid $( \bar { x } , \bar { y } ) \text { of } D \text { are } \bar { x } = \frac { 1 } { 2 A } \oint _ { C } x ^ { 2 } d y , \bar { y } = - \frac { 1 } { 2 A } \oint _ { C } y ^ { 2 } d x$ where A is the area of D. Find the centroid of the triangle with vertices (0, 0) , ( $4$ , 0) and (0, $8$ ) .

A) $\left( \frac { 4 } { 3 } , - \frac { 8 } { 3 } \right)$
B) $\left( \frac { 1 } { 8 } , \frac { 1 } { 8 } \right)$
C) $\left( \frac { 1 } { 8 } , \frac { 1 } { 64 } \right)$
D) $\left( \frac { 1 } { 8 } , \frac { 1 } { 4 } \right)$
E) $\left( \frac { 8 } { 3 } , \frac { 8 } { 3 } \right)$