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. . Oxy D - , D f(x, y).
D n D1, D2, D3, , Dn, ( ). s(Di) Di; diam(D) , D:
;
d Di:.
Di (i = 1,2, , n) Pi = (xi, yi), f(Pi) = f (xi, yi), .
, D Di, Pi, f(x, y) D, f(x, y) D .
f(P) P, ds ds = dx dy, :. , ,.
.
. f(x, y), g(x, y) D, D, .
-. . , .
. D D1 D2, , .
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D :.
-: , .. , Pi. , .
. : , - Di f(Pi); - , .. ( , Di, f(Pi)). , , D, - z = f(x, y), , D, Oz. .
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12. .
.1. f (x,y) D, y=x y=x 2, ( (5) (7)).
17 |
. 17 D. (5), (7), .. D , , , , .
(5), .. y, x , x. D x = 0 x = 1, , 0 £ x £ 1. y, : x Î(0,1) , . D , (. 17). y = x 2, y = x, .. x 2 £ y £ x. ,
D = {(x, y): 0 £ x £ 1, x 2 £ y £ x }. , (5), :
(7). x, y , y D y = 0 y = 1, , 0 £ y £ 1. , x, y Î(0,1) , . .. B 1 D x = y, C 1 D , x y .
D = {(x, y): 0£ y £1, y £ x £ }. , (7),
2.
. 1), D, . D 1 - , D 2- . .. x, , D 1 D 2 . D 1 0 ≤ y ≤ 1, 0 ≤ x≤ , .. D 1 = {(x, y): 0 ≤ y ≤ 1, 0 ≤ x≤ }. , D 2 1 ≤ y ≤ e, ln y ≤ x≤ 1, .. D 2 = {(x, y): 1 ≤ y ≤ e, ln y ≤ x≤ 1}. , D = D 1È D 2 (.18). D x = 0, x = 1 y = x 2, y = . , D = {(x, y): 0 ≤ x ≤ 1, x 2 ≤ y≤ }. (5)
18 19 |
3. I = .
13. . .
. . , .
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x = rcosα
y = rsinα
? r , r = √(x² + y²) ≥ 0, α , , 2π.
, .
, .
ΔN = ΔφΔρ:
:
, :
|OCD|2 = (Δφ/360)π(ρ + Δρ)²
|OAB|2 = (Δφ/360)πρ²
:
|ABCD|2 = (πΔφ/360)((ρ + Δρ)² ρ²) = (πΔφ/360)(2ρΔρ + Δρ²) = πΔφ2ρΔρ/360 + πΔφΔρ²/360
, . :
πΔφ2ρΔρ/360 + πΔφΔρ²/360 = πΔφ2ρΔρ/360 = ρΔφΔρ
ΔN = ρΔM
, ρ, . , , ρ. , .
14. .
, , , ().
.
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n- :
F(x, yn, yn-1, y, y)=0
n- f(x), (a,b) n , F(x,f(x), f(x), fn(x))=0
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15. .
16. , .
17. - . . .
f 1(x), f 2(x), , fn (x) , c 1, c 2, , cn, , , . .
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