set-up-an-integral-for-the-volume-of-the-solid-obtained-by-rotating-the-region-bounded-by-the-given-

Question

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ y = e^{-x^2} $ , $ y = 0 $ , $ x = -1 $ , $ x = 1 $

(a) About the x-axis
(b) About $ y = -1 $

Wen Zheng · Accepted Answer

the problem is set up an integral for the volume after stolid, obtained by rotating the region, bounded by the Cuban curves about the specified the line. Then use your calculator to wire into the integral. Corrected too fast is the more places how to et out X axis Had to be about why you could you next. What? The first two weeks catch the graph of this. Read it The region is this part. So if we rotated this region about X axis, we have four inch axe cross section Is that disk on? His blindness is cultural. You too negative X squared then half. Why is he called Tau Pi? Into girl From Nick to one to one into negative X. Claire squired Jax Since this is the coat too. Three point seven above and too far O b the tent about why you could connect it. Why for each acts that can see the cross section is a washer. And in your lioness is culture one on outer ideas is a cultural. Each connect you X square past one says volume is cultural pie into girl from nick to want one each connective x square. Ah, Swan squared minus one. Jax, this is culture. The king&#x27;s point one for you. Three one two

Chris Trentman · Answer

it&#x27;s were given a region founded by curves and a line and were asked to set up an integral for the volume of the solid obtained by rotating this region bounded by these curves about this line Then rescues are calculated to evaluate the integral correct to five decimal places. The curves are y equals x squared and x squared plus y squared equals one. Oh yes where why is greater than or equal to zero? This is going to be The Parabola and the upper half of a circle of Radius one. And in part a the line is about the X axis like that. Mhm. Black Cat December 22 Central we were playing HFS&#x27;s Yeah we&#x27;re putting up opening for jimmy&#x27;s chicken sack, Coachella. And so we&#x27;re essentially rotating a thin vertical strip. Do A D. C. Did part about the X axis. We get a washer in other words with inner radius various X squared and an outer radius. Yeah. Well if you solve the equation for the circle and take why to be positive, this is the positive square root of one minus X squared. Oh, your will and grace. So we use the washer method and the volume V is pi times the integral from uh the okay, X coordinate of the intersection of Y equals x squared and x squared plus Y squared equals one. We&#x27;ll we&#x27;ll do this first actually. So when are these intersecting? Well, this is when Y plus. Okay. Just taking chest john great equals one. So solving for why we get Mhm. What? Well, we really don&#x27;t want to find why we want to find X. So instead, yeah, we have X squared plus X squared squared equals one. So we have X to the fourth plus X squared minus one equals zero. And solving we get that X squared is equal to negative one plus or minus the square root of one plus four. All over to This is negative one Plus or -15/2. However, X squared is positive. Seriously considering one plus Route five you too and therefore X is going to be plus or minus the square root of route five minus 1/2. And if you have to calculate this is approximately point 786 Yeah, that fits Okay, so our volume V is pi times the integral from our left point of intersection which is negative square root of route five minus 1/2 2. Positive square root of route five minus 1/2. The of our outer radius squared. Mhm. This is the square root of one minus X squared minus our inner radius X squared squared d x £5000. And so simplifying this is pi times the integral from negative square root of route five minus 1/2 Square root of route 5 -1/2 of this is one minus X squared. Mhm. Yeah. They all do that voice. Yeah. Sorry this woman is x squared this is one minus X squared minus X. To the fourth dx. This is the exact volume. However if you plug this into a calculator This is approximately .354 four. Mhm. Yeah. Always running. Sorry. 3.54459. Yes. Yes. Dude, I love it. Yeah. When they even approach a joke Yeah. Six. Which if we&#x27;re rounding 25 decimal places. This is 3.54460. Mhm. Of Oh no. Mhm. They&#x27;re like I like like all right so that&#x27;s part a in part B. Were asked to use the same curves but now we&#x27;re going to reflect a rotation, say about the y axis. Yes that way. In the Greater Fort Wayne. Yeah. Now we&#x27;re rotating a thin vertical strip about the Y axis. So we get a cylindrical shell. It says Veterans first response with the radius of X. Those flags and the height publishers which is you solve the circle square root of one minus X squared minus X squared Yeah. Try to carry. Therefore our volume is equal to using the shell method. The integral from 02 x equals our X intercept are right. Most X intercept This was the square root of route 5 -1/2 mm. This is the same as from part A of two pi times our radius which is X times are height which is the square root of one minus X squared minus X squared dx kids. Yeah. For me. Mhm. Yeah, for America. Mhm. So this is the exact value of our volume, right? However you can use a calculator to find an approximate volume. If you do. The volume is approximately among trying to extract kind of branches weighing down his shoulders. He&#x27;s just yeah. Yeah. Most Spy comes .9999. Mhm. Eight. Yeah. Two. Yeah. That would be from yeah. Yeah. Sure. Is there a guy who that&#x27;s his? Yeah. Have a sturdy base your friend couldn&#x27;t have. He would be your deal. Last list.

Rebecca Polasek · Answer

So if I want to find value, then I need to know the area and rotate it. So first, Well, my crocs here x squared and then actually, the circle equation X squared plus y squared equals one. So the area bounded by these two equations. Is this part here? I&#x27;m shaking. Now. I need to know where I&#x27;m revolving around in this problem. Let&#x27;s revolving around the X X is so if we&#x27;re roaming around the X axis. Noticed all this missing piece here from the volume, which means it&#x27;s a washer set up now because it&#x27;s the washer set up. We know that volume is pi times the integral R squared minus r squared. We use DX because it&#x27;s the X axis, which means I also need X for my limits of the integral. So x one and X two are gonna be the bounds there. Now, the other thing that&#x27;s important or helpful is I noticed that this has symmetry. So personally, I would like to just use this portion of the shaded area and then double my set up. So I&#x27;m actually going Teoh be solving two pi. I&#x27;m gonna go from zero to wherever they meet here, so I still need to find this X is my ending limit. And then still the same stuff R squared minus R squared DX. It just makes it easier especially. But I am working out my hand to use a zero there for the lower bound on the limit. And this is a time when it&#x27;s great to use a calculator because we&#x27;re gonna get to decimals here but again, on opportunity to make it simpler work by using that zero bound. Now, if I want to see what this looks like in the graphing calculator I would want to solve for Y, and solving for y equals you would have y equals plus or minus one minus Sorry, square root of one minus X squared, right? You subtract the X squared over first, and then you square root both sides and because you square, where did you get? Plus or minus? The negative only defines this lower part of the circle, so I actually don&#x27;t need the lower part of the circle for my volume. So really, it&#x27;s just the positive version of the square root. So as I&#x27;m graphing it, I&#x27;m just going to see this top curve which didn&#x27;t see the top part of the circle, which is good, because that&#x27;s what I need. I would recommend using graphing calculator to find this intersection point so I would graph X squared, and I would grab the square root of one minus X squared to find where those two graphs are equal to find that X to value. So if I use the intersection of my graph to solve this, I see that X is let me tell you exactly what I got. 0.786151 I may go to several decimal places, so that way, at the end, I can actually still list my answer to five decimal places. So if I want my answer to be five decimal places, I would like for my work to go to at least six. Help me keep it exact. So, feeling it, this is my upper bound on the integral. So it goes from zero toe 00.786151 Now I need to find capital R squared and lower case R squared. So capital R squared is what if this volume were all the way filled in? Nothing was missing. That would mean I would shade everything going from the top curve down to my access of revolution. Well, the tub curve is defined by the circle. The circle is the square root of one minus X squared, and then access of revolution is your X axis. So that&#x27;s just a minus zero. Not super helpful as I&#x27;m solving through that part of it, right? Doesn&#x27;t really define anything there, so I don&#x27;t have to write the minus. Seagram, lower case R is what are we taking away? Well, we&#x27;re taking away all this missing stuff below the shaded area until we get to the axis. So below the shaded area, notice is defined by your parabola down to the access. So actually X squared down to zero. Okay, X squared down to zero. I could just right as X squared. Now, the in time this integral is really awesome to rewrite because you have a square root that&#x27;s gonna cancel out. And then you just have that X squared squared, which becomes an extra the fourth so pretty easy to rewrite the integral part of us. But then maybe not a simple Teoh evaluate that crazy decimal. So you could go one step further to get this anti derivative. Or at this point, you could type this whole thing into the calculator. So if I were to take that whole thing into my calculator multiplying by two Pi and Brian the Integral I get 3.544 5912345 decimal places. That is a decimal approximation of my volume here for the graphs between that X squared in the circle function going around the X axis.

Carson Merrill · Answer

So for this problem were given some bounds for integration Y equals zero y equals cosine squared X. You know X is between negative pi over two and pi over two and we want to do it about the X axis. So what we end up getting As a result, if we have our graph here, the shaded region is going to look like this. And then again, that&#x27;s going about the X axis. So it&#x27;s going to give us this solid. Um, and when we since this is going to be a disk, we&#x27;re going to use the disk formula, so it&#x27;ll be volume pie. Um, our bounds are negative. Pi over 22 pi over two and then our radius in this case is cosine squared X. So then we square that and we have derivative or the integral. So when we take the inter row of this, our area is going to end up being two pi times zero to pi over two of the courtesan to the fourth X DX. We can use our calculator here and what we end up getting is that we have a volume of 3.70110 then for part B. What we want is to do it about, um y equals one. So as a result, we&#x27;re gonna end up getting a washer. So to do this washer, we&#x27;re gonna use the washer formula. Remember, the washer formula is pi times the bounds times the honor roll with bounds A and B are one squared minus R two squared d x So now what we end up having is our volume being equal to hi. Going from negative pi over two two pi over, Tim, Our first radius is one. So one squared minus one minus cosine squared, X squared DX. We can simplify this further, and what we end up getting is pie. Actually, we can do to pie good from zero two pi over two times of to cosign squared X minus co sign to the fourth X. Now you can differentiate this and put it in our calculator. What we end up getting is for our volume 6.16 850 And remember, these answers are going to be in units cubed since we&#x27;re dealing with volume

Victoria Dollar · Answer

Okay. So for this problem, we are asked to set up a integral for the volume where it&#x27;s rotating about the boundaries provided and then calculate the integral up to five decimal places. So we have that. Why equals zero. Why equals co sign Squared X and then we have between negative pi over two at, uh and pi over two for X. So the first part we want to do is across the x axis. So what we want to do is we want to set up our, um are too great. And so are we going to use the disk method for this? And so what we want to do is we want to set up our volume thio equal pi along the integral beat from B to A or A to B R squared DX. So what I want to do is my are is actually going to be my co sign Squared X. So what I&#x27;m going to do is it&#x27;s going to be pie from pi over two to negative pi over two for are integral and then cosine squared X squared D X. And then, of course, this is going to be pie from pi over two to negative pi over two and then it&#x27;s gonna be cosign to the fourth x dx. So what I can also do since it&#x27;s going to be symmetrical because it&#x27;s a cosine value. What we want to do is go ahead and set this up to be two pi and then either choose negative pi over 2 to 0 or zero to pi over two. I prefer dealing with positives, so co sign for X dx and then by utilizing a calculator, we can find out that this is going to be three pi squared over eight, which is approximately 3.70110 And then, of course, for part B were asked to do this across. Um, it&#x27;s why equals one. And so what we want to do is use the very we only use a washer method for this one. So just a zey refresher. The washer method is volume equals pie from a to B of our one squared minus are two squared dx. And so since we&#x27;re now dealing with y equals one, that&#x27;s gonna be our our one. So we&#x27;re gonna have pie from, uh, oops from, uh, pi over two to negative pi over two, and we&#x27;re gonna have one squared minus cosine squared, X squared DX. So what we&#x27;re going to do for this is it&#x27;s gonna be pi from negative to positive pi over two. And now we&#x27;re going to square this, so it&#x27;s going to end up being one minus and then we I&#x27;m sorry, this is actually supposed to be one minus cosine squared X because if we think about the graph, we&#x27;re gonna have our cosign, and then we&#x27;re gonna have one. So our coastlines kind of kind of look something like that. So we&#x27;re gonna have to have that difference, but from one to co sign squared, and then we&#x27;re gonna have one minus that. So we&#x27;re gonna have one minus two cosine squared X plus cosign to the fourth ex D X. And then, of course, are ones air going to cancel. So we&#x27;re gonna have one minus one, which will be zero, and then we have negative. So it could be to cosign squared X minus co signed to the fourth Ex D X. And then So then again, we want to do something similar to what we did with the last one and do two pi from pi over 2 to 0 of to cosign squared X minus cosine to the fourth ex DX. And then if we utilize a calculator, it will tell us that the answer is five. Pi squared over eight, which is approximately 6.16850

Amrita Bhasin · Answer

we know that we need to first find the intersection points, which means that if acts is two acts over one plus x cubed again, we&#x27;re setting. If wise acts and wise to x over one plus x cube, we&#x27;re setting the why equal to the y in terms of X. So for X, we end up with acts. Times X cubed minus one is zero and factored form, which gives us two values for ax zero and one. That&#x27;s good. The&#x27;s There are two bounds. Now we know the radius of each cylinder is X minus negative one, which is the same thing as X plus one. We know the height is gonna be two acts over one plus X cubed minus acts. Which means now we know we can write the integral as two pi times integral from 01 of X plus one against our times H So we&#x27;re literally just putting it. This is R H is two acts over one plus X cubed minus axe. So just our terms. It&#x27;s just these two values plugged in times. Divac&#x27;s okay, this was part A was right. The interval part B is putting into your calculator. We&#x27;re gonna put it into a graphing calculator. On we get 2.36164 and you can plug in exactly what we just wrote. If you have a graphing calculator is the same thing as two pi times two pi over three spurt of three minus five Sex But you can literally just plug in this whole integral into your calculator.

Wen Zheng · Answer

So for this problem, 31 for part A. We want to look at the rotation about the X axis. Um, So what we&#x27;re gonna end up getting is a disk with radius E to the negative X squared. So our volume since the disk is going to be, um, the integral from negative 1 to 1 of pi r squared. But in this case, our radius is e to the negative x squared, squared. So we have our d X, then we simply integrate. So what we&#x27;re gonna integrate is negative 1 to 1 pi e to the negative two x squared d x, Um and we can use a calculator or anything like that to find this and what we end up getting is that this volume is approximately 3.75 83 and that&#x27;s going to be in Units Cube. Because it&#x27;s a volume, then what we&#x27;re finding, since it actually has to be to five decimal places, we&#x27;ll give this 2 to 5. So it&#x27;s our five decimal places. And now for part B, we want to do it about the y axis, so it&#x27;s going to be similar, although this time it&#x27;s going to be a washer instead of a disk. Um, so our volume is going to be a little bit more complicated. It&#x27;ll be pi times the interval, um, high times the inner role from negative 1 to 1 of negative one plus one plus e to the negative x squared, squared DX. We can calculate this out further, and what we end up getting is approximately. The volume is 13 point 14 And you can do this in your calculators yourselves. And once again, remember that some units cute.

Problem

Set up an integral for the volume of the solid ob…

Problem 31 Medium Difficulty

Answer

a) $V \approx 3.75825$
b) $V=13.14$

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a) $V \approx 3.75825$b) $V=13.14$

Calculus

Grace H.

Heather Z.

Kayleah T.

Samuel H.

Integration Review - Intro

Definite vs Indefinite

James StewartCalculus

Grace H.

Heather Z.

Kayleah T.

Samuel H.

a) $V \approx 3.75825$
b) $V=13.14$

James Stewart

Calculus