Please use integration to find the center of mass of the triangle `(0,0)(2,0)(0,8)`

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Let O(0,0) ,A(2,0) and B(0,8) be the vertices of triangle OAB.

equation of side AB of triangle OAB is

y-0={(8-0)/(0-2)}(x-2)

y=-4(x-2)

y=-4x+8

Thus are of the triangle OAB is

`Delta=int_0^2ydx`

`=int_0^2(-4x+8)dx`

`=(-2x^2+8x)_0^2`

`=(-8+16)`

`= 8` sq.unit

Thus centre of the mass

`barx=(1/Delta)int_0^2xydx`

`=(1/8)int_0^2x(-4x+8)dx`

`=(1/8)(-4x^3/3+4x^2)_0^2`

`=(1/8)(-32/3+16)=2/3`

`bary=(1/Delta)int_0^2 (1/2)y^2dx`

`=(1/16)int_0^2(-4x+8)^2dx`

`=1/16int_0^2(16x^2-64x+64)dx`

`=1/16{16x^3/3-32x^2+64x}_0^2`

`=(1/16){(16xx8)/3-128+128}`

`=8/3`

Thus centre of mass of the triangle is`(barx,bary)=(2/3,8/3)`

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