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(0) Find the imnge under [email protected] Je of tle rectanglee and the region enclosed by the rectangle with vertices nt 0, J j Justify Yo MawCTFind the Iqage Mmdf(:) Lop? of the...

Question

(0) Find the imnge under [email protected] Je of tle rectanglee and the region enclosed by the rectangle with vertices nt 0, J j Justify Yo MawCTFind the Iqage Mmdf(:) Lop? of the Iegion the sexolid qurant bouncled

(0) Find the imnge under [email protected] Je of tle rectanglee and the region enclosed by the rectangle with vertices nt 0, J j Justify Yo MawCT Find the Iqage Mmd f(:) Lop? of the Iegion the sexolid qurant bouncled



Answers

Let $S$ be the boundary surface of the box enclosed by the planes $x=0, x=2, y=0, y=4, z=0,$ and $z=6$ .
Approximate $\iint_{S} e^{-0.1(x+y+z)} d S$ by using a Riemann sum as in Definition $1,$ taking the patchcs $S_{i j}$ to be the rectangles that are the faces of the box $S$ and the points $P_{i j}^{*}$ to be the centers of the rectangles.

Hello, everyone. Today they're going to solve the problem of us your function of this eggs that day, please. Three X squared. What is quality plus Explain. Okay. A vector field and see the boundary of the rectangle in the figure using the stove stare on was integral have dot They are It was double integral all sigma, then close f Don't nd it's with Jessica's no brainer Or are they cross f not? Then the deep it so we'll signifies that surface broken by the close to cook Production of the surface on the X Y plane is are is from ex commonly that is from zero less than a week or two x less than or equal smart zero less than a record by less than a record toe three, then dill Cross. It will be Bill Cross f will be I until burrito. It's except did do. But all right, three x square by square. Okay, nobody does it. Excellent. So we get like X I plus X minus Lee the plus six x y square kick. So deep off Excellent said will be zed minus like which is they Jeep, which is my energy. Let's keep So Bill crossed and the then deep is it called toe? Why? Minus X plus 66 way squid For using the result of the surface integral, they get closed. Integral have dark d r equals 0 to 1 and from 0 to 3 y minus X plus 66 white square The way BX, which is equals 0 to 1 y squared by two planus. Exley. Plus it looks like you from 0 to 1 DX. So here it's like sarah 02 So here it's, like 0 to 1 nine by two. Plus 51 x Biggs, which is 92 x minus driven by two exist squared 0 to 1, which is 111. Their stand off a question. Thank you.

Were given a boundary surface and the function and rest to approximate the surface integral of dysfunction along this boundary surface by using a riemann some and in this three months, some we're going to take the patches to be rectangles that are the faces of the boundary surface and the points p i. J star to be the centers of the rectangle. So our surface s This is the boundary surface of the box and closed by the planes. X equals zero X equals two. Why equals zero? Why equals two certain way equals four Z equals zero NZ equals six and our function is he to the negative 0.1 times x plus y plus z. So, first of all the faces of the box and the planes X equals zero and X equals two. They have a surface area of four times six or 24 each, and the faces have centers which are well, we have zero y equals two z equals three and X equals two y equals two z equals three. Now the faces in the plains y equals zero and why equals or yeah, have a surface area which is two times six or 12 and they have centers one zero three and one, four three and finally faces in the plains. Z equals zero and Z equals six have a surface area of two times four or eight each. Yeah, and have centers of X equals one y equals two z equals zero and 12 six. Now, as were instructed for each face, we're going to take the point p i j star be the center of the face. And of course, our function F. As I pointed out before, this is e to the negative 0.1 times x plus y plus z And so, by definition, one we have that the surface integral of f over s be approximated as the remond, some with terms we have f of zero 23 So this corresponds to function at the center of the plane of these surface in plain X equals zero times the area that face which is 24 plus the function evaluated at the center of the face corresponding to plain X equals two times the area that face plus you function evaluated at the point, the center of the face corresponding to the plane y equals zero so f of 10 three times the area of this face, which is 12 plus function evaluated at the center of the face corresponding to y equals four, which is one 43 times the area of this face which is 12 plus the function evaluated at the center of the face corresponding to the plane z equals zero which is one 20 times the area of this face which is eight plus the function evaluated at the center of the face corresponding to the plane z equals six which is 126 times the area of this face which is eight and plugging this in the 24 times. In fact, during this is each of the negative 0.5 plus e to the negative 0.7 plus 12 times each of the negative 0.4 plus e to the negative 0.8 plus eight times each of the negative 0.3 plus e to the negative 0.9 and plugging us all into a calculator. This is approximately equal to 49.9

So use all Christians in the valley with this light grow. So by current serum, this will be the seventh thie area. And to grow off the ripped half of this with respect to X, so to eat with X minus the river till this was respectful. Why? Which is eaten with X, uh, are about all right, ta here. So either the ex d a. So this will be isa rocked. M go from zero zero to three zero three four to the oh four. So is simply X from zero to three. Wife from zero to four on the beach with X the the X from zero to three. Wife zero two four. So if it was ex from zero to three So we have, uh, these three minus one d Why And this is a constant always simply have the muted prized by four


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