f[x_]:=(x+1)^2+1
g[x_]:=-4x+9
Plot[{f[x],g[x]},{x,-1,2}, Filling-> {1->{2}}]

Integrate[g[x]-f[x], {x, 0, 1}]
(11/3)

h[x_]:=x^3
j[x_]:=2-x^2
Plot[{h[x],j[x]},{x,-1,2}]

Integrate[h[x], {x, 0, 1}]

N[Integrate[j[x],{x, 1, Sqrt[2]}]]

N[1/4+0.2189514164974602]

Integrate[j[x],{x, 1, Sqrt[2]}]

N[-(2/3)-(3/4) + (2/3)*(2^(3/2))]