In the first section of this note,we deduce,with the aid of theapproximation theorem for an integrable function,the following result from atheorem of Wilkins concerning Maréchal's integration scheme:If G(x)∈L(a,b)and if F(x,X,Y)is continuons on α≦x≦b,|X|≦1,|Y|≦1,then(1)(?)F(x,cosλx,sinλx)G(x)dx=1/2π(?)F(x,cosφ,sinφ)G(x)dx dφ.The Riemann-Lebesgue theorem is a special case with F(x,cosφ,sinφ)=cosφor sinφ.In particular,if(?)where S(z) is analytic inside,and on the unit circle |z|=1,except at afinite number,of singularities in |z|<1,then an application of the residuetheorem gives (2)(?)F(cosλx,sinλx)G(x)dx=i∑R_k·(?)G(x)dx,where ΣR_k is the sum of residues of S(z)inside |z|=1.In the second section,we consider a type of integration which approxi-mates a double intehral over a region D,where D is bounded by continuouscurves y=f_1(x),y=f_2(x)and lines x=α,x=b.Under general conditions itis shown that(3)(?)F(x,g_λ(x))h_λ(x)dx=∫∫_D F(x,y)dx dy,where g_λ(x)=cos~2(λx)f_1(x)+sin~2(λx)f_2(x),h_λ(x)=(f_1(x)-f_2(x))|sin(2λx)|,f_1(x)≥f_2(x),(a≤x≤b).