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a. Let $B$ be a cylindrical shell with inner radius $a$ outer radius $b,$ and height $c,$ where $0<a<b$and $c>0 .$ Assume that a function $F$ defined on $B$ can be expressed in cylindrical coordinates as$F(x, y, z)=f(r)+h(z),$ where $f$ and $h$ are differentiable functions. If $\int_{a}^{b} \tilde{f}(r) d r=0 \quad$ and $\tilde{h}(0)=0, \quad$ where $\tilde{f}$ and $\tilde{h}$ are antiderivatives of $f$ and $h,$ respectively, show that $$\iiint_{B} F(x, y, z) d V=2 \pi c(b \tilde{f}(b)-a \tilde{f}(a))+\pi\left(b^{2}-a^{2}\right) \tilde{h}(c)$$b. Use the previous result to show that$$\iiint_{B}\left(z+\sin \sqrt{x^{2}+y^{2}}\right) d x d y d z=6 \pi^{2}(\pi-2)$$where $B$ is a cylindrical shell with inner radius $\pi,$ outer radius $2 \pi,$ and height 2

a) $\pi\left(b^{2}-a^{2}\right) h^{\sim}(c)+2 \pi c\left(b f^{\sim}(b)-a f^{\sim}(a)\right)$b) 6$\pi^{2}(\pi-2)$

Calculus 2 / BC

Calculus 3

Chapter 5

Multiple Integration

Section 5

Triple Integrals in Cylindrical and Spherical Coordinates

Integration Techniques

Multiple Integrals

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Lectures

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(a) find the spherical coo…

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were given, Um, problem where we have a cylindrical show. So something that looks if we were to graph it like this with the inside kind of out of it. This is a big old hole in the middle. And we're told that for this show, the, um inside radius is given by a The outside radius is given by B and then the high other. The head of the shell is given by Sea Knight C and we're told that we can You can write the function f of X y Z in the form lower case f of our plus h of the were here when we're talking about our, um and see these air these air cylindrical coordinates. So those were the normal formulas were also given that if we have F Tilda is the anti derivative, um, well, say I've till the end age. Still, though, sorry, F Tilda Age Tilda are anti derivatives of f and H, respectively. And using that week it we have some information here specifically that the integral from A to B the same they MB that we have here for the inner and outer radius of the anti derivative of f with respect to our d. R is zero and then we also have that the ah anti derivative of H zero is equal to zero. So we're going to do with this is we're going to find a formula for the integral, the triple integral of B. We're going be here is this, um, cylindrical shell of f of x y z um Devi and we're gonna use the information given to us to find to find a formula for this. Okay, so the first thing we can dio as we can we can change this to cylindrical coordinates. So if we do that, then we can write our triple integral should actually pretty easy. We go from 0 to 5. That would be our data. Our radius it goes, since we're looking at the region between and be then goes from A to B and then the value of Z goes from zero. And since the height of see we're going to see from here, we know that we can rewrite our function f in terms of our NZ and so we could do that. But when we change to cylindrical coordinates, we also have to multiply by our it's actually would look right at this one. Well, multiply each of these individually by our our time f of R plus R Times agency and then we go backwards in the order of the Integral. So the first or the inside Integral is for Z Super easy. And then we put d r and then data. I know we're just gonna were gonna calculate this, and we're gonna use the given information here about the anti derivatives to work this out. So the first thing we can do is we'll do the integral we respect dizzy and that integral. So we have from 0 to 2 pi from A to B. And then so the entire derivative of that is going to be our FFR Z, um, plus R and then the anti derivative of H. Whatever. That is that we don't know. They don't give us unless that's fine. We don't need it from zero to see if the Argies data. So by plugging in, we see we get since equal, Let's go down here we have the integral from zero to pi in a row from a to B of. So the first thing we have in our half of our time. See, um plus R h prime or age Still TFC. Now we're weak. We're gonna attract in playing NZ are zero. But if we plug in zero into the first term here than our f of our time, zero gives zero. And then we're also told sexually what we can write out our fr zero. But they were also told that we have our h Tilda of zero here h 200 We were given up here. Zero. So, actually, so both of these are going to cancel it DRD thing. Okay, so now now we can do is we can actually simplify this one step further. Um, since none of these are in terms of data, we can actually write this as the interval from 0 to 2 pi times One of data times the interval from A to B, our FFR our time. See, plus our h primacy d r. We can do this and it's helpful because now we see that this is just a single variable for our and the integral From 0 to 51 it is just gonna equal to pi. So So now where should you aware about this integral here and then multiply that by 25 and we're done. So in this case, um, we could we're gonna Yeah, we're gonna break this apart. It'll be easier to just look at two separate pieces. So the first part would be from a to B of We'll take the sea out there are fr d r plus. And then h h told O. C again is is actually a constant So you can take that out age tilt of C in a row from a to B of our d. R. So when we when we work it out this way, Um, well, we can see is that in this first inner girl, it's not necessary obvious what the first step is. But we're told information about the anti derivative of death, and so that would lead us to think that this might be a good integration by parts problem. Um, because we can take the derivative, are get rid of our and we actually know something about the anti derivative F so that even though we don't know what efforts would still use the entire derivative. So we're doing a great effort to get you vehicles are Do you, Nicholas D. R. We would say Devi equals f of r d R and therefore V is just the entire derivative of f of our so that when we were at that integral. So if c times, um, we have our times f tilda of our minus the anti derivative of just f told of our but we're evaluating these from a to B. So this is for me to be here, and this is for me to be here. So again, this is only talking about this first inner girl here. Now we can see that this is going extremely helpful because again, the given information the anti derivative or the integral the anti derivative from A to B is zero. So you can actually grow in here. We can cancel that out. And so now we're left with is just plugging in Vienna into our it's affecting them. So here we get C times B. In this case, f Tilda be minus c times a f. Tilda a van. Okay, so that's that piece right there. Uh, and then we have over here. If we look at this inner girl, the antidote artists are screwed over too. So this is going to be hte Tilda of C times In this case, B squared over two minus a squared over two. And now we can put all together. So here we were, adding, so we should bring We should be careful here that, um that this is plus this right here. Tilt of C a times B squared over two minus x squared over two on that. We need to remember that in the end, we were multiplying by. This is two pi right there. So then our final answer would be two pi times bcf, Tilda of B minus a C F. Tilda of a plus age Tilda. See, and be screwed over two minus a squared over two. Now this this doesn't match exactly what? What's the answer given? But we can see that, um, by the shooting in the two pi into so we can consider, like, to this to be the first part on this to be the second part so we can distribute that into both of those parts. But then we can also factor out to sea in this in this 1st 1 and so we can see if we, you know, multiply this all out. We get to pie Bacon Factory to see from both of those terms and we're left with B after all, to be minus f till day plus And if we multiply the two pi and then the 2nd 1 since both these air being divided supposed to be squared in a sort of being divided by two multiplied by two will cancel those out. So we're just left with pi h told of C and then just b squared minus eight square. And that is the final answer for for part A for part B, we're told, or we're asked to use this result to show the following. So for part B, we need to show then if we integrate over the ball, be of Z plus sign square root of X squared plus y squared dx dy wind Easy that this actually this equals six Hi square times pi minus two. Now we can see here that, um that the sea here is like the age of the and then given the extra plus y squared, that sign is the f of our okay. It's definitely going way changes since the cylindrical coordinates. We would get this as a function of our It always did. He was just check to make sure that, um, everything all the properties matchup. So the first thing we see here we have to check is that the anti derivative of H at zero gets a serum, which is definitely true because the anti derivative of H is just c squared off to plug ins. Eerie gets here. We also need to check that the integral Oh, sorry. We also need to see what B is here. Let's do them. B in this case is given to be, um, the cylindrical shell. Well, just draw the picture with inside radius of pie, so it will look like this, or we cut out the middle. So the inside radius is pie. The outside radius is two pi on the height of it is to, so we need to check them. So, um, you need to check that the integral in this case from pie to two pi of the anti derivative of f of our equals zero. Well, let's see your the said so f of Oregon is just sign of our the anti derivative of that is negative coastline of our And then when we do the integral from pie too high of negative coastline of our what we're gonna get is we're gonna get negative Sign of two pi plus sign of pie. And the important part here is that a sign of two pints on a pie are both zero. So this is just here, which is what we want to be. So this problem satisfies all the requirements. So now we just we just plug everything in and we hopefully get a six pi squared times pi minus two. Okay, so let me Let's rewrite the formula down here. So the formula we have for this dancer here should be to pie time. See, B f. Tilda of B I miss a f tilda of a plus pi b squared minus a squared times H Tilda of scene. Okay, so So the answer for integral then our triple integral that we have be for Z plus sign of squared of X squared plus y squared dx dy y dizzy will equal to pi. In this case, C was the height which is to see us to hear. So a is pie V is two pi I see it too. So 25 times Two times two pi times f tilda of be So in this case, that is negative. Co sign of two pi minus pi times f tilda of a. So this is negative co sign of pie plus pi times. Let's see here. Be squares off. Four pi squared minus pi squared. And then h prime of C h h Matilda is ah playing to We get four squared over two, such as to. And this should this should be our our ah, answer that we got up here, So let's simplify some things here. So we have two pi. Times two is four time. Um, Ms Missing a pregnancy there. Okay, um and then we have in the inside here. So co signer, Two pies. One. So this is negative One. So it's negative to buy there. Uh, coastline of pie is negative one. And so then that's become positive. One times pot are positive on times. Negative pie. Is that negative pie? Plus, we have pie times. Three pi squared. We'll put this the two there. So we have 25 there. So there we have four pi times negative. 35 plus six pie to the third. If we keep on going, we get negative. 12 pi squared plus six pie to the third. And now, Now we can see that we'll get the answer they want. Um, if you look at the given answer here, six pi squared up. I minus two. If we factor out six pie from this right here, we get exactly that. We have 65 um, squared and then we're left in the inside. Pie minus two. That's answer.

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