V2=∫S2(x)= a1(b-b1) /2h*(h-x)+ (a-a1) (b-b1) (h-x)2/2h2
F(h)-F(0)= a1(b-b1) /2h*h2/2+(a-a1) (b-b1) /2h2*(h3/3)= a1(b-b1)h/4+(a-a1) (b-b1) h/6
因此,
V=V1+2*V2=(a1+a)*h*b1/2+ a1(b-b1)h/2+(a-a1) (b-b1) h/3
=3(a1+a)*h*b1/6+ 3a1(b-b1)h/6+2(a-a1) (b-b1) h/6
=h/6*[3(a1+a) *b1+3a1(b-b1)+ 2(a-a1) (b-b1)]
= h/6*[(a1+a) (2b1+b)+a(b-b1)]
= h/6*(2ab+ab1+a1b+2a1b1)
得到结果。