第十二章重积分 xfx 证明:x(/≥x(2D]e →∫x(x/(x2x(x(xk oxf(x)/2(kody()f(dxdy e∫jy(()-y2()/()≥20 00 -ef(x)/20)-y'ov)/(x)xdy 20 ∫(y(x)()-yf(o)/(x)d ∫()f()-xf(x)() x(D≥x(2)e ((x)f2(y)-yf()/(x)+y(y)f()-xf2(x)()ay 因:x(x)2()-yf2(v)(x)=f(x)/(v)x/(v)-yf(x) 则,x/(x)f(y)-y2()/(x)+y/()/2(x)-xf2(x)/() =/(x)()(x-yf(x)-f(v)≥0 10.若Wx∈[0<m≤f(x)≤M,证明: M+m dxdy≤ 0≤ysl 第十二章重积分第十二章 重积分 第十二章 重积分 证明: ( ,0,1) ( ,0,1) 2 x f  x f  ( ) ( ) ( ) ( )      1 0 2 1 0 2 1 0 1 0 f x dx xf x dx f x dx xf x dx  ( ) ( ) ( ) ( )      1 0 1 0 2 1 0 2 1 0 xf x dx f x dx xf x dx f x dx  xf(x)f (y)dxdy yf (y)f (x)dxdy    1 0 1 0 2 1 0 1 0 2  ( ( ) ( ) ( ) ( )) 0 1 0 1 0 2 2 −   xf x f y yf y f x dxdy  ( ( ) ( ) ( ) ( )) 0 1 0 1 0 2 2 −   xf x f y yf y f x dxdy ( ( ) ( )− ( ) ( )) =  xf x f y yf y f x dxdy 1 0 1 0 2 2 (yf (y)f (x) xf (x)f (y))dxdy  − 1 0 1 0 2 2 ( ,0,1) ( ,0,1) 2 x f  x f  ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) 0 1 0 1 0 2 2 2 2 − + −   x f x f y yf y f x yf y f x x f x f y dxdy 因： x f(x)f (y)− yf (y)f (x) = f (x)f (y)(x f(y)− yf (x)) 2 2 则， x f(x)f (y) yf (y)f (x) yf (y)f (x) x f (x)f (y) 2 2 2 2 − + − = f (x)f (y)(x − y)(f (x)− f (y))  0 10.若 x0,1, 0  m  f (x)  M , 证明： ( ) ( ) ( ) M m M m dxdy f y f x y x 4 1 2 0 1 0 1 +       