Question

Вычислить интеграл . Срочно

Answer 1

$\large \int\int_{D}(18x^2y^2+32x^3y^3)\mathrm{dxdy}=2\int_{0}^{1}x^2\mathrm{dx}\int_{-\sqrt{x}}^{x^3}(9y^2+16xy^3)\mathrm{dy}=***\\ \int_{-\sqrt{x}}^{x^3}(9y^2+16xy^3)\mathrm{dy}=3y^3|_{-\sqrt{x}}^{x^3}+4xy^4|_{-\sqrt{x}}^{x^3}=3(x^9+\sqrt{x^3})+4x(x^{12}-x^2)=3x^9+3\sqrt{x^3}+4x^{13}-4x^3\\ ***=2\int_{0}^{1}(3x^{11}+3x^2\sqrt{x^3}+4x^{15}-4x^5)\mathrm{dx}=2\int_{0}^{1}(3x^{11}+3\sqrt{x^7}+4x^{15}-4x^5)\mathrm{dx}={x^{16}+x^{12}\over2}|_{0}^{1}+{4(\sqrt{x^9}-x^6)\over3}|_{0}^{1}={1\over2}(1-0+1-0)+{4\over3}(1-0-1+0)={1\over2}\cdot2=1$