Область горизонтального входа
-1 ≤ y ≤ 1
0 ≤ x ≤ y^2
[m] ∫ ∫ _{D}(x^2+y^2)dxdy= ∫^{1}_{-1}( ∫^{1}_{y^2}(x^2+y^2)d)dy=[/m]
[m] =∫^{1}_{-1}((x^3/3)+y^2x)|^{x=1}_{x=y^2}dy=[/m]
[m]=∫^{1}_{-1}((1/3)+y^2-((y^2)^3/3)+y^2\cdot y^2))dy=[/m]
[m]=∫^{1}_{-1}((1/3)+y^2-(y^6/3)-y^4)dy=((1/3)y+(y^3/3)-(y^7/21)-(y^5/5))|^{1}_{-1}=[/m]
[m](1/3)\cdot (1-(-1)+(1/3)\cdot (1^3-(-1)^3)-(1/21)\cdot (1^7-(-1)^7-(1/5)(1^5-(-1)^5))=(2/3)+(2/3)-(2/21)-(2/5)=(26/21)-(2/5)=...[/m]
2 способ.
Область вертикального входа
0 ≤ x ≤ 1
-sqrt(x) ≤ y ≤ sqrt(x)
m] ∫ ∫ _{D}(x^2+y^2)dxdy= ∫^{1}_{0}( ∫^{\sqrt{x}}_{-\sqrt{x}}(x^2+y^2)dy)dx=[/m]