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Use thc Divergence Theoremn evaluate NdS where F(x,Y, 2) = x+yj zk and S is thc surface of the cube ((1,Y, 2) : s* $ 2,0 <y < 2,0 < z < 2}.(c) 32None of...

Question

Use thc Divergence Theoremn evaluate NdS where F(x,Y, 2) = x+yj zk and S is thc surface of the cube ((1,Y, 2) : s* $ 2,0 <y < 2,0 < z < 2}.(c) 32None of these

Use thc Divergence Theoremn evaluate NdS where F(x,Y, 2) = x+yj zk and S is thc surface of the cube ((1,Y, 2) : s* $ 2,0 <y < 2,0 < z < 2}. (c) 32 None of these



Answers

Use the divergence theorem to evaluate $\iint_{S} \mathbf{F} \cdot d \mathbf{S}$ where $\mathbf{F}(x, y, z)=y^{2} z \mathbf{i}+y^{3} \mathbf{j}+x z \mathbf{k}$ and $S$ is the boundary of the cube defined by $-1 \leq x \leq 1,-1 \leq y \leq 1,$ and $0 \leq z \leq 2$.

Hi problem for 46 using the divergence serum, we got the triple integral or volume integral of three X squared Y plus four TV. Setting this up we go from 0 to 2 photo one Make it 4 to 1 three x squared y plus four dx dy y DZ Simplifying this integral. We get three times e from 0 to 2 times Why squared over two from 0 to 1 plus for Z from 0 to 2. Why 01 x from record for no one and this comes out to one for 16 or 17.

Our problem. Fortunately, using the divergence serum we're on Evaluate thesis er fis integral. Alright. So looking at this function when we find the Grady int, it's it's obviously equal toe one we can see ah d d x of x y That's gonna be why then, um, do you want the ax of, um, one half white square that becomes wise So wise y zero and then d d d z of z is one so ingredient FFs to go toe one. So there just be one times the triple integral Devi That's equal to the volume. So now we just can't get the volume of the of Per Probably Lloyd, which is gonna be one half times pi times one squared times four minus one And then at that two pi r squared times one, it comes out to 5/2 pi

Are using the divergence serum for problem for 47 sending it for Integral. Gonna look like that. Then we plug in our limits. So from 0 to 2 pi 0 to 2 here to rid Z of Z squared. R T R T z t data simplifying okay and then simplifying this integral we get that comes out to three times Seda from 0 to 2 pi, you eat the four There were four from 0 to 2, the articles 24 5.

My problem for 09 He's a divergence hearing to find m r. He squared TV all rights except the triple under Groll. All right, so the diversion serums gonna give us one third zero Come zero comma the cute dot Okay, simplifying this using spherical get Costa into the fourth fight. Yes, I've been simplifying this integral. We got one third too high. My zero of negative negative one minus 1/5. And that comes out to four pi over a 15.


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