Calculate the integral ∬S𝐅· d 𝐒, where S is the surface of...

Calculate the integral $\iint_{S} \mathbf{F} \cdot d \mathbf{S}$, where $S$ is the surface of the half-ball $x^{2}+y^{2}+z^{2} \leq 1, z \geq 0$, and $\mathbf{F}=\left(x+3 y^{5}\right) \mathbf{i}+(y+10 x z) \mathrm{j}+(z-x y) \mathbf{k}$

Transcript

00:01 For this problem let e be the set of points x comma y comma z such that x squared plus y squared plus z squared is less than or equal to one where z is greater than or equal to zero so with this pretty much we are going to have the upper half of a of the unit sphere in fact this is a solid so we're going to call this a ball.

00:33 So here we're going to have the x and the y plane otherwise known as the plane z equals zero.

00:40 So here we are going to have the x -axis coming out this way, the y -axis coming out that way, and the z -axis is going straight through our hemisphere.

00:50 So in our case, notice that theta will range from 0 to 2 pi, while fee ranges from 0 to pi over 2, and row ranges between the values of 0 and 1 so in our case by the divergence theorem by the divergence let's call it capital d for divergence theorem theorem the double integral over s of f dotted with ds also known as the flocks is equal to the triple integral over the object e of the divergence of the vector field f dv so in our case we are giving that the vector field f is equal to the vector field whose components are x plus 3 y to the fifth powered comma y plus 10 y plus 10 um x z and the last component for this let me make room for the last component here the last component is simply z what about the divergence of this vector field the divergence of f is just going to equal to 1 plus 1 plus 1 which is equal to 3 so therefore we are going to have the triple integral over e of 3dv so become that is the triple integral over e of 3dv so we can pull the 3 outside of the triple integral that's just going to be 3 times the triple integral over the object e of 1dv by definition, this part right here is known as the volume or volume of e.

02:44 So since we have a hemisphere, instead of having four thirds, we're going to have two thirds times pi r cube, where the radius of our sphere is 1...

Question

Calculate the integral $\iint_{S} \mathbf{F} \cdot d \mathbf{S}$, where $S$ is the surface of the half-ball $x^{2}+y^{2}+z^{2} \leq 1, z \geq 0$, and $\mathbf{F}=\left(x+3 y^{5}\right) \mathbf{i}+(y+10 x z) \mathrm{j}+(z-x y) \mathbf{k}$

Please give Ace some feedback