y=e⁻ˣfor x∈[0,1].
A=∫e⁻ˣdx is area under the curve (=total mass), so A=(-e⁻ˣ)[0,1]=1-1/e.
Theory: Rectangular strips width dx, height y, area=ydx (=point mass);
Centre of mass of strip is at half its height (½y).
x-moment is xydx; y-moment is (½y)(ydx)=½y²dx.
x̄=∫xe⁻ˣdx/∫e⁻ˣdx, ȳ=½∫e⁻²ˣdx/∫e⁻ˣdx for x∈[0,1] in all integrals.
Integration by parts:
Let u=x (du=dx) and dv=e⁻ˣdx (v=-e⁻ˣ),
∫xe⁻ˣdx=uv-∫vdu=-xe⁻ˣ+∫e⁻ˣdx=-xe⁻ˣ-e⁻ˣ.
x̄=-(xe⁻ˣ+e⁻ˣ)[0,1]/(1-1/e)=(1-2/e)/(1-1/e)=(e-2)/(e-1)=0.4180.
ȳ=(-¼e⁻²ˣ)[0,1]/(1-1/e)=[(e²-1)/(4e²)]/[(e-1)/e]=(e+1)/(4e)=0.3420.