# Find the center of mass of the region bounded between `y=-x^2-x+4` and `y=-2x^2-2x+5`

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### 1 Answer

green is graph for curve y=-2x^2-2x+5 and red is for y=-x^2-x+4

These two curves are inter secting each other when x=.618 and x=-1.618.

Thus area of the region bounded between these curves is

`Delta=int_-1.618^.618(-2x^2-2x+5+x^2+x-4)dx`

`=int_-1.618^.618(-x^2-x+1)dx`

`=(-x^3/3-x^2/2+x)_-1.618^.618`

`=(-.078-.191+.618)-(1.412-1.309-1.618)`

`=1.864`

Let `(barx,bary)` be coordinate of centre of mass of bounded region

`barx=(1/Delta)int_-1.618^.618 x(-x^2-x+1)dx`

`=(1/1.864)(-x^4/4-x^3/3+x^2/2)_-1.618^.618`

`=.536{.076-1.008}`

`=-.50`

`bary=(1/Delta)int_-1.618^.618 (1/2)((-2x^2-2x+5)^2-(-x^2-x+4)^2)dx`

`=(1/Delta)int_-1.618^.618(3/2)(x^4+2x^3-3x^2-4x+3)dx`

`=(3/(2Delta))(x^5/5+2x^4/4-3x^3/3-4x^2/2+3x)_-1.618^.618`

`=.80{.945+4.645)`

`=4.447`

Thus

`(barx,bary)=(-.50,4.447)`