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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 $F(x, y, z)=f(r) g(\theta) h(z), \quad$ where $f, g,$ and $h$ are differentiable functions. If $\int_{a}^{b} \tilde{f}(r) d r=0$ where $\tilde{f}$ is an antiderivative of $f,$ show that$$\underset{B}{\mathscr{M}} F(x, y, z) d V=[b \tilde{f}(b)-a \tilde{f}(a)][\tilde{g}(2 \pi)-\tilde{g}(0)][\tilde{h}(c)-\tilde{h}(0)]$$where $\tilde{g}$ and $\tilde{h}$ are antiderivatives of $g$ and $h$ respectively.b. Use the previous result to show that$$\iint_{B} z \sin \sqrt{x^{2}+y^{2}} d x d y d z=-12 \pi^{2}, \quad \text { where } B$$is a cylindrical shell with inner radius $\pi,$ outerradius $2 \pi,$ and height $2 .$

a) $\left(\int_{a}^{b} f^{\sim}(r) d r=0\right)$b) $-12 \pi^{2}$

Calculus 2 / BC

Calculus 3

Chapter 5

Multiple Integration

Section 5

Triple Integrals in Cylindrical and Spherical Coordinates

Integration Techniques

Multiple Integrals

Missouri State University

Harvey Mudd College

Baylor University

Boston College

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So we are given a cylindrical sell shell Be so slick, something like this and we are asked to integrate over the shell where we're told the inner radius of the shell is given by a The outer radius of the show is given by B on the high of the shell is given by scene. We're told that the function f of X y Z can actually be rewritten in cylindrical coordinates, Um, as f ar as a multiplication of functions f of our GF Data and H of Z were also given for the information that the integral from A to B of the anti derivative. So again that f till there is the anti derivative of F um, that that anti derivative is zero or that the integral a zero. And so we're going to use this information to find the formula for the integral of for B of the function f of X y z. So the first thing we're gonna do here as we're going Teoh you know, change change this into, um, cylindrical coordinates. Since we know we can write it in this this nice fashion. So if you look at our cylindrical shell we have, um dangle goes from 0 to 5. Data goes from 0 to 5. The radius. Um, since we're looking at the inside of the shell is kind of the region we're looking at here. The radius goes from A to B where a was the inside to the outside shell and then the height of it s C, which means we're going rz value goes from zero to see. And now we can rewrite our function f in the cylindrical coordinates. You know, f of our GF ada h a z de. In this case, we go opposite of the intervals. So we the inside integral is Enciso do DZ every two d r then we to death data. But also remember to multiply when we're changing to cylindrical coordinates most five yr. Well, now what we can do is we can actually break this all apart so we can rewrite this as, um the integral from 0 to 2 pi Uh, in this case, geese Ada di fada, times the integral from a to B of our FFR d r and then the in a row from zero to see of HFC do you see? And we can now work this out pretty nicely. Um, the first in the last one. The first interim last integral will be easy Teoh to find in terms of the anti derivative. So we don't actually know what the formulas for the specific anti derivative. But we're going. We're going right, um G Tilden, aged Children to represent those. So let's focus on the middle integral here, since this one's going to cost us the most grief. Now what we know is that we know that the integral of just the anti derivative by itself zero and so, looking at what we're given here, it would make sense to you something like integration by parts. Because here we are. We're multiplying by our in the driver of that will go away, becomes one And then we know stuff about the anti derivative f. So let's try that we have u equals r Do you hold on equal d are just what we wanted and then we have DV is the f of our d. R and V is now just the DEA anti derivative of us. Whatever that maybe we don't know doesn't matter. So in this case, what we can do is wait. Gonna rewrite this integral as our Tilda of our from A to B minus the integral from A to b of just f. Tilda of our But we're giving up here that this integral equals zero. So that just canceled out. So now we can actually do is you can actually weaken right? Our final And here because the 1st 1 So this interval here is just going to equal G. Tilda of two pi minus g. Tilda of zero, where we don't know anything else. We don't have to. This is just the fundamental theme. A calculus. We know. Now this middle one is going to be B f. Tilda. Be minus a. I've told of a And then we know this last one here, just like the 1st 1 will be H. Tilda of C minus H. Tilda of zero. Where again H Tilda is the anti derivative of H so and so the answer is multiplying. All three This together our final antihero, v g Tilda two pi minus gee Tilda zero times b f tilde key minus a. After that, a, uh, times h told o c minus age, Tilda zero. And that's exactly what we want. Now we have part a for part B. We're going to now use this results to actually to find a mineral. So let's say we have the triple integral over the domain be of Z sign of square root off X squared plus y squared dx dy y DZ. We're going to show that this is going to equal negative 12 pi square. Now, what we're told about B is that it's a cylindrical shell, just like we had, um, previous previously, um, and were given the inner shell in the outer shell. In this case, the inner shell is just pie. The outer shell is to pine on the height of it is to. So for this problem, a equals pi vehicles to pie and C equals two. So we have to confirm a few things here. The first thing we need to confirm, um, is that this could be ruin in India in the format where I'm multiplying, you know, the function of the function of our and function if they did together. But we can see again that that's true, because here the Z is the HFC the sign, even though it doesn't look like it square next group was twice. Court is Are So this is like our f of our, um where you could just repeat this a sign of our and there is no theater. But if you don't want one, imagine that there's were most find by one there, then the one is our GF data. So this could be written in that form. But the second thing we knew the check. So if we have f of our equal sign of are we all seem to check that the integral of the anti derivative from A to B zero. So the anti derivative here is negative coast out of our and so we're doing the in a role in this case from from A to B is pi to pie. Since this is the specific values have here of negative co sign our d r. But when we worked that out, we know the anti derivative of coastline is just just signed. Um, since I took the negative outside the integral, So it's negative. Sign of two pi minus sign of pie. But signer 25 cent pie are both zero. So that is your just like it's supposed to be. So this fits all the requirements that we need, um, to show this. And so now, now we can just use the formula we have up here. So let's let's write these out. So we have h of C is C, which means H tilt of Z is C squared over two. So there's our H and H still down. Here's our F and F dildo. And so then g Zeta is equal to one set me into Ji Tilda of Fada is equal to theta and now we're gonna plug those in with our values of A B and C. So then the answer for the integral is going to be So we have g told of two pilot, which is just two pi minus G Tilda zero, which is just zero. So that times now we have B, which is two pi times f tilda of B. So I've told as negative co science negative co sign of two pi minus a which is pi times negative co sign of a, which in this case is still pie. And then that times we have two squared over two. So that's H h tilda of to minus zero squared over two. So simplifying this, we get to pie. Negative or co signer two fires one. So it's just negative. One negative to pie. There co sign of pious negative one times a negative, is positive, has a negative pie. It's just negative pie. And then, um, off See zero script over 202 Squared over two issues, too. And so now he gets two times. Um, well, let's let's simplify this year. So this is really for pie on the front. So just to five times to and the name of 25 minus pi is negative three pie and now we get the answer of negative 12 pi squared, which is exactly what we were looking for up here. They told vice squad, and so we're done.

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If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

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