Question

Let 𝐹 = 2𝑥𝑧𝑖 − 𝑥𝑗 + 𝑦 2𝑘. Evaluate 𝑉 𝐹𝑑𝑉 is the region bounded by the surfaces 𝑥 = 0, 𝑦 = 0, 𝑧 = 𝑥 2 , 𝑧 = 4.

Answer 1

$F=2xzi-xj+y^2k$

The region bounded by the surfaces: $x=0,y=0,z=x^2+y^2,z=4$

(Modification $z=x^2+y^2$ was made, because the region is bounded in y-direction only from one side: y=0.)

$divF=\frac{\partial}{\partial x}(2xz)+\frac{\partial}{\partial y}(-x)+\frac{\partial}{\partial z}(y^2)=2z$

Let the region bounded by the surfaces: $x=0,y=0,z=x^2+y^2,z=4$

$\iiint divFdV=2\int^2_0dx\int^2_0dy\int^4_{x^2+y^2}zdz=\int^2_0dx\int^2_0(16-(x^2+y^2)^2)dy=$

$=\int^2_0(16y-x^4y-2x^2y^3/3-y^5/5)|^2_0dx=\int^2_0(32-2x^4-16x^2/3-32/5)dx=$

$=(32x-2x^5/5-16x^3/9-32x/5)|^2_0=64-64/5-128/9-64/5=$

$=1088/45$