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vector calculus was never my strongsuit, but here is the formula set for centers of mass of 3d regions (rotating a curve creates a 3d region W):
xbar = ( tripleint[W] x p(x,y,z) dxdydz ) / mass,
ybar = ( tripleint[W] y p(x,y,z) dxdydz ) / mass,
zbar = ( tripleint[W] z p(x,y,z) dxdydz ) / mass, where
mass = tripleint[W] p(x,y,z) dxdydz, where
p(x,y,z) is the mass density. in your case, i'm not sure if e^(-x) is the mass density or its rotation the region, but i suspect it's the mass density.
this is all from Vector Calculus 4th ed by Marsden & Tromba.
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