91影视

The given integral is $鈭�\mathrm{鈭�}脳Vd蟿$.

Divergence theorem, often known as Gauss' theorem or Ostrogradsky's theorem, is a theorem that connects the flow of a vector field across a closed surface to the field's divergence in the volume enclosed. According to this theorem, the surface integral of a vector field over a closed surface, also known as the flux through the surface, equals the volume integral of the divergence over the region inside the surface.

Solve the equation as shown below.

$\mathrm{鈭�}脳\mathrm{V}=\left(3x^2鈭�2x\right)y+\left(3y^2鈭�4y+1\right)x+2z$

$=3x^{2}y+3x{y}^2鈭�6xy+x+2z$

Then, the integrals become as shown below.

${鈭�}_0^1鈥�{鈭�}_0^1鈥�{鈭�}_0^1鈥�\left(3x^{2}y+3x{y}^2鈭�6xy+x+2z\right)dxdydz$

${鈭�}_0^1鈥�{鈭�}_0^1鈥�\left(y+\frac{3y^2}{2}鈭�3y+\frac{1}{2}+2z\right)dydz$

${鈭�}_0^1鈥�\left(\frac{1}{2}+\frac{1}{2}鈭�\frac{3}{2}+\frac{1}{2}+2z\right)dz=1$

$鈭�\mathrm{鈭�}脳Vd蟿=1$

Hence, the solution of the integrals is $鈭�\mathrm{鈭�}脳Vd蟿=1$.