Let R Be a Plane Region of Area a Bounded $( \bar { x } , \bar { y } )$

question 43

Short Answer

Let R be a plane region of area A bounded by a piecewise-smooth simple closed curve C. Using Green's Theorem, it can be shown that the centroid of R is $( \bar { x } , \bar { y } )$ , where $\bar { x } = \frac { 1 } { 2 A } \oint _ { C } x ^ { 2 } d y$ $\bar { y } = - \frac { 1 } { 2 A } \oint _ { C } y ^ { 2 } d x$ Use these results to find the centroid of the given region.
The triangle with vertices $( 0,0 )$ , $( 3,0 )$ , and $( 3,4 )$ .

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