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Numberade F.

Calculus 3

4 months, 3 weeks ago

So this is our surface integral. We need to evaluate. So we need to use the divergence theorem. First of all, what is the region? The region is the region bounded by the parable Lloyd's. That is going to X squared plus Y squared. And the plane, is it difficult to put And obviously the boundary of the surface will be circling. So you can see this diagram. So this is the diagram, this is the parable or is it is equal to X squared plus Y squared. This is my plane. That is equal to four. So if you can see this is a circle, X squared plus Y is critical before. So it's better to use cylindrical coordinates to evaluate design divergence theorem. So what does divergence theorem says is the surface integral of a vector field? Dot ds D sigma is volume integral of the divergence of the vector field and the volume element D. V. So this is the whole human race of the triple integral. So let's use cylindrical coordinates according to cylindrical coordinates We have accessible to our cost theater. Why are scientists and that is equal to that itself? And the jacobean of the cylindrical coordinates is well known to. First, I'm not doing that division to jacobins are. So that means first we need to set up the limits of the integration of the triple integral. And also we need to calculate the divergence of F. So first let's calculate the divergence of F. So this is my vector field. So the divergence of F. The vector field and dr office going to the X component I call it as a fun divided by the X. So this is my F of X Y square. This is my F two. This is my everything. Let's do F two by the way. Why Let's do F three by doesn't So what is the F one x 2? X is by square? What is do F two by Dubai? It is extra square. And what is the It is lucy Now when we substitute the cylindrical coordinates, the divergence becomes X squared plus y squared. In R squared enthusiasts as it is music. All right. So now so let us substituting now and if you see the limits of integration, Zet is depending on our because the parable Lloyd, Z is equal to X squared plus y squared. So the limits of our that belongs to 02. Our square I'll use malted. Okay, I'll use small it's a small thing so small. That belongs to Ask her to 0 to ask her ask her to fall Because art is going from 0 to I will be going from 0-2 region. The boundaries a circle. The bounty is a circle of really spoke. So the equation of this circle is expert. Was very square is for the maximum radius is too the minimum radius is zero but that is actually depending on r squared. So that will be taking values from our square to four. So that is aspect for and r is zero to go. Of course 3 to 0 to two points because we have a complete revolution. So now we are ready to calculate them Surface integral. So the surface integral as after our basic model as he could do 3-0-2 pi Are 0-2 That are square to 4th. The divergence. How much we got? The divergence is R squared is 20. Let's go ahead. Let's do that to the Jacoby in art. D R. D data. So this is the order of integration suppose let us integrate with respect to them. Integration of that is equal to r squared forward, husband and wife is argue to reduce that integration of our cube with respect to Z. Is our cubes and integration of I can do this in order to do that. I'll introduce you. What is the decoration of Brazilian? Does that square So it is ours that square. And the limits are low limit is sorry low limit is our square upper limit is four. So let's substitute both the limits So it becomes our Cube into four. Art into 16 minus bracket are Cuban to our square. Our into our scoreboard square that is our powerful. So basically we'll get for our cue for our cube plus 16 R minus r par fives are perfect. You are for five. So that means now let us spending your winter vision. It is theta is equal 0-5. R is equal to 0- two. For our cube Plus 16 R -2 are .5 D. R. Did he talk this integration? You can split it for R. And Theta because for the independent in degrees So theta is equal 02-5 digital into I is equal 0-2. For our cube Plus 16 R -2 are for five D. R. This is two because integration of the data is data And Parliament is zero. This integration let us do What is integration of hierarchy? Powerful integration of six. R 68 R squared integration into our power five. It is our power six divided by three substitute. It's Caltech 2.2. Lower limit is anyway zero upper limit letters substitute to bar for 68 into 218-4 32 By 64 divided by three. So that will be 2.2 48 -64 divided by three. It is two pi by three and 2 18. So there's 1 60 by by. So the answer for this surface intravenous 1 16 x 13.

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