练习22(1) 二。2 ∥=44 y-型: ≤x≤√y,1≤y≤2 =式 2 x-型:1≤y≤2x2,√2 x≤1 01 2 x2≤y≤2,1≤x≤√2 =d∫/(x,y)d 5i dxh f(x, y)dy+hdx-2f(x,y)dy 2 K心
练习2.2(1) 二. 2. o x y 2 y = x 2 y = 2x 21 1 2 y −型: , 1 y 2 2 x y y x −型: 1 21 x 1 x 2 1 2 , 2 y x2, 2 x y = yy I dy f x y dx 2 21 ( , ) = + 2 2 1 21 121 2 2 ( , ) ( , ) x x dx f x y dy dx f x y dy
三.3 xt y x+y=1 x十 J 1+x y 十dr x+y e-e K心
三. 3. o x y x + y = 1 x − y = 1 − x + y = 1 x + y = − 1 + − − + − = xx x y I dx e dy 1 1 0 1 −− + + x x x y dx e dy 1 1 10 . − 1 = e − e
四.记D:a≤x≤b,a≤y≤b b f∫(x)d dx=lf(x)dx f() df()a 小y f(x) a f(r) b b b 1 f(x)x∫ d=f(y)小y F(y)dxdy af(r) a f(r) D(r) b b 1 f(y) f(r) dxdy af(r) 2 D ∫(y),f(x)≥】2akd=(b-a2 2DL∫(x)f(y) K心
四. 记D : a x b,a y b. = = D b a b a b a b a dxdy f y f x dy f y dx f x dx f x f x dx ( ) ( ) ( ) 1 ( ) ( ) 1 ( ) = = D b a b a b a b a dxdy f x f y dx f x dx f y dy f x f x dx ( ) ( ) ( ) 1 ( ) ( ) 1 ( ) = + D D b a b a dxdy f y f x dxdy f x f y dx f x f x dx ( ) ( ) ( ) ( ) 2 1 ( ) 1 ( ) = + D dxdy f y f x f x f y ( ) ( ) ( ) ( ) 2 1 D 2dxdy 2 1 ( ) . 2 = b − a
五.记F(x)=nf(x),则F(x)=f(x) F(1)=n∫(x)dx=A,F(0)=0, Sdxf f(x)f(ydy=5o f(x dxf f(y)dy Sof()F(1)-F(x)ldx F(ilof()dx Jof(x)F(x)dx =42-51P(x)F(x)t=42-1F(x)dF(x) =A2-F2(x)= (1) 2 A A2 2 K心
五. ( ) ( ) , 0 = x 记F x f x dx 则F(x) = f (x), (1) ( ) , 1 0 F = f x dx = A F(0) = 0, = 1 1 0 1 1 0 ( ) ( ) ( ) ( ) x x dx f x f y dy f x dx f y dy = − 1 0 f (x)[F(1) F(x)]dx = − 1 0 1 0 F(1) f (x)dx f (x)F(x)dx = − 1 0 2 A F (x)F(x)dx = − 1 0 2 A F(x)dF(x) 1 0 2 2 ( ) 2 1 = A − F x (1) 2 2 1 2 = A − F . 2 1 2 2 1 2 2 = A − A = A
练习22(2) 0 元 6≤ 元 2 2 I=∫(x2+p2)d-∫(x2+y2 正方形 半圆形 , dx(x2+y2)dy-J2 d0 f /2rdr 2 K心
练习2.2(2) 三. 2. o x y1− 1 − 2 = + − + 正方形 半圆形 I (x y )dxdy (x y )dxdy 2 2 2 2 − − = + 11 0 2 2 2 dx (x y )dy − 10 2 232 d r rdr 四. o x y . 2 2 −