Задача 69719 ...
Условие
∫∫_(D) (12xy+9x^2y^2) dxdy
D: ...
математика ВУЗ
236
Решение
★
0 ≤ x ≤ 1
-x^2 ≤ y ≤ sqrt(x)
[m] ∫∫_{D}(12xy+9x^2y^2)dxdy= ∫ ^{1}_{0}( ∫^{\sqrt{x}} _{-x^2}(12xy+9x^2y^2)dy)dx=∫ ^{1}_{0}(12x\frac{y^2}{2}+9x^2\frac{y^3}{3})|^{\sqrt{x}} _{-x^2})dx=[/m]
[m]=∫ ^{1}_{0}((12x\frac{(\sqrt{x})^2}{2}+9x^2\frac{(\sqrt{x})^3}{3})[/m]-[m](12x\frac{(-x^2)^2}{2}+9x^2\frac{(-x^2)^3}{3}))dx=∫ ^{1}_{0}(6x^2+3x^{\frac{7}{2}}-6x^5+3x^8)dx=[/m]
[m]=(6\frac{x^3}{3}+3\frac{x^{\frac{7}{2}+1}}{\frac{7}{2}+1}-6\frac{x^6}{6}+3\frac{x^9}{9})|^{1}_{0}=[/m]
