solve-the-given-problems-by-integration-find-the-volume-generated-when-the-first-quadrant-area-bou-3

Question

Solve the given problems by integration.
Find the volume generated when the first-quadrant area bounded by $y=e^{x}$ and $x=2$ is rotated about the $x$ -axis. [Hint: After setting up the integral, rewrite $\left.\left(e^{x}ight)^{2} 	ext { as }\left(e^{x}ight)\left(e^{x}ight) .ight]$

Audrey Fong · Accepted Answer

in this question, we want to find the volume generated by this curve And bounded by as equals to two and rotated about the excesses. In general when we have a cough White was two epics. And we want to find the volume from Mexico to A to be generated about the excesses over here. We&#x27;ll just get a small strip over here. The week is so small. We&#x27;ll just call it D. X. Okay. And of course the heart here is why now as he generates over the excesses is creating a circular disk. The circular, jeez the wife is the X. Or the height of the disease dX. And the radius is why. So the volume of this. This is pie our square heat, right pi y square the X. No. To get the entire volume from A to B. We would just sum up now since this is a continuous submission. So it will be an integration sign from A to B. Hi y square the X. You can leave the pie outside. And so that would be your volume. So to find a volume for this so volume equals two pi summing from 0 to 2. Why square the X. Yeah. So that will be E. To the X times X. Because that&#x27;s why square E. X. There are a couple of ways to integrate this. But for this topic we&#x27;re using substitution method and integration power rule. So for substitution method licks you&#x27;ll be eating power X. The U. Over the X. Is it a power X. So the you is it apart X. Dx. So this part over here is the you now we need to substitute these X values here as well. We used values. So when X equals to the lower limit zero You is either power zero and there is one When X is the upper limit of two. U.S.E. No, we also need the power rule. The power rule says that when we integrate you to power and we respect to you, we will just get you to the power and plus one. So we add one to the power and divide by the new power. Now I put classy here because this is an indefinite integral but for definite integral we do not need to put A plus E. So now we are ready to substitute soap. I now instead of X equals zero, it will be you close to one and for the upper limit of X equals two, you&#x27;ll be U equals to the square. Now it&#x27;ll power X over here is you and it&#x27;ll power X dx over here is the you. So now we&#x27;re ready to integrate, this is you to power one. Using Power rule, it will become you to power one plus one over the new power one plus one and is within the epitome of e square and lower the MIr one mm. Now I can bring this two Outside so it will be pi over two. Yeah, you square. So you square. I&#x27;ll be subbing the upper limit in first, so will be e square square- and serving the lower limit in so B -1 square. So the final answer is hi over to It to about 4 -1 units. Cute.

Wen Zheng · Answer

problem is calculated. The Walliams generated a pirate trading the region under the by the group&#x27;s Why could you almanacs y zero and actually come to about each? Access is about y Axis B is about X axis. First week in jars, a graph. Why just there&#x27;s a graph of the curve. Why onto our next? Why could zero, two, two. So the region is this part now for a If we rotated this written about y axis, then we can use cylindrical shall my thirty to find it. It&#x27;s a warrior as he coached you. Chew pie into girl from wants you to ex Toms on X Jax Then the kind used my theory of integration my powers, not you. That&#x27;s the next. And there be a promise they caught you axe. Then you promise he goes to one of our acts and obviously could want half X squared. So this is the coat too. Too high times you can&#x27;t sway. So this is one half X squared. Alan likes from one to two minus the integral from one to two. Your prime comes ways that this is one half x x. This is culture to pi times flying two one one two Dysfunctions. This is two times now and two minus Cyril and minus one force. I square from one to two. This is too high times. She would have into minus this isthe three over more for being obtaining DeVry. It&#x27;s a region about axe access. We can use this comm answered. Yes, integral from one to two pi times. Alan likes squire yaks thiss come either then forthis into girl we can light. You can also use integration. My parts that you is equal to our end likes squire on being one. Then you promise to two times out on Max over Max. No problem. And obviously, But now this integral as goto pi times you comes, please, that this is X and X corner from wan t to minus in general, I wanted to you prom. Have, please. This is to Tom&#x27;s out on Knox Jax. Now for this integral, we can use intuition apart again. But you got your our necks. We promise you could want. Then you promise they put one. Iraq&#x27;s be able to act. This is the cultural pie times looking to and want to dysfunction. This&#x27;s too now end too squire. Minus two times is the integration by parts. This is X our necks from one to two minus and to grow from. I wanted to you Crumb Tab. Sleep this one. Max Zico. Hi to Pam&#x27;s out one too. Choir, Linus. Two times to have our end too. Minus NCIS. Why? This&#x27;s the result.

Vikash Ranjan · Answer

fine. So lame. Okay, West, be Waas 11 2 The sun radius is the distance from X to the line as judo less. I&#x27;m getting the next exit. And then do then Sam? Yes. Is he? No. The same is the distance begin. Duh. Why the line you bore? Its there is so all right comes beautiful Rex. The volume is on one. Hey, what is even as to buy it off, you know and do. And then my ex Indu, you didn&#x27;t borrow the s by, you know? And then do your parents. There&#x27;s my still buy. You know I do. That&#x27;s your deals. Is it good enough? Do my under Riggins. It is our X Julia minus Dubai. And your parents minus Rex. It is you. So this is the guns to buy and do the bar Do minus. Do minus by left will go on and do minus Do minus. D&#x27;oh! My party. This is England given as to buy and then do in to do my ass. And do you mind if I do? And then you in to do my less to class? Once this is in bloom. Bye and do minus by two minus one given by do minus four on and do glass. So it is even as bye to buy one mass. This is the one, man, spit it The answer.

J Hardin · Answer

this problem is from Chapter seven section to problem number sixty two in the book Calculus Early Transcendental Sze in Tradition by James Door Here we like to find the volume obtained by rotating the region bounded by the curves Why equal sign Square necks y equals zero in between X equals zero and pi and we&#x27;d like to rotate this about the X axis. It&#x27;s over here on the right. You have a rough sketch of the graph y equals sine squared and blue for X between zero and pi and we also have and read the horizontal line Y equals zero. So from these graphs we can see that the area that we&#x27;re interested in rotating is this area between the blue and the red which weaken to know by our. So after we rotate this thing about the X axis, we have the following another of sketch and here&#x27;s a cross section and let&#x27;s do another cross section. So what do you see? That our our cross sections or washers arm sees me disc. So we&#x27;d like to find the volume which will be the general of the area of the cross sections. So we have to be more precise here since we obtained the volume of the solid By letting the washers moving the extraction, we&#x27;LL have to integrate with respect to X So X is going between zero and pi. Now we need a formula for a in terms of X so is equal to pi r squared because is a disc a circle including the inside of the circle So we have pi r squared area of a circle Now we need to represent our in terms of X. So in Ackley case, we could come back to the diagram that we have and this is one of our cross sections in the center. That&#x27;s a circle to find the radius. Let&#x27;s go to the center right here on the x axis and to find the radius. We see it&#x27;s just a distance from the center, which is why I called zero up into the graph. So we have r is equal to why the length of our here. Why am I not zero? But since why is on the graph sign next clear, we can replace why with sine squared legs. So we have a equals pi r squared, which is pie science squared X Square, and that equals pi. Science of the fourth Power X. So this is the term that will be integrating. And let&#x27;s pull up that pie outside the integral sign. So picking up where we left off, we have the equals pie in a girl&#x27;s ear. Opie Science. If we have science to the fourth power, let&#x27;s go ahead and re write that as science. Where Time Sign square. Then we can use one of the path a grand entities to rewrite us. So let&#x27;s use the identity. That&#x27;s deeds. Science for Lex is one minus two, called Cho San to X, all divided by two. So using this formula, we have one minus co. Sign to X over one minus costs and to accept it, too. So let&#x27;s go ahead and pull this four out of the denominator and simplified the numerator as much as we can. So this true evaluating the numerator, multiplying this out. We have one minus to co sign to X plus co sign squared to X. Now we can help. We can integrate one and co sign of two works to integrate co sign squared of two X. We should use the pathetic and identity, which says co sign Squared X is one plus cause I to x opportune. We have whatever for in a girl&#x27;s ear in a pie one minus two co signed to X And now using this identity over here on the right, we can write this as one half plus co sign of two times two x so co sign up for X over two. And now we can go out and agree each of these terms. And also note that this one in the one half well combined to three. House What? So what&#x27;s going on? Integrate this we have power for and a girl of three over two. Three except you and a girl of negative to co sign two works is minus two. Signed to X over, too. An integral co sign up for X is signed for X over four. So we have four times two and we have the end points see, or a pie. So it&#x27;s Gordon. Come down here to the side of plug. All this in by Over four now was plugging pipe burst three point two, minus sign of two by plus sign of a bi over eight And then when we plug in zero, we see that each of these three terms go to zero. So we subtract nothing, and we could see from the unit circle Sign of two pi zero sign of a pi zero. Some were left with Pyla for times three pi over too. So again, three pies square all over eight and there&#x27;s our area. That&#x27;s a volume.

Bobby Barnes · Answer

if we want to find the volume of the solid revolution formed by rotating X axis with the region after Becks is equal to X to the X or two times E to the why is he going zero exclusive Negative to excise the good one. We end up warning so fine. So we have this area here and blue and then we want to rotate this. So look, something kind of like this here when we will use thie solid of revolutions method So remember that we want So our ball you we&#x27;ll be equal to pi Times are bounds on are acceptable So the negative two one And then we have a full backs squared and then we integrate with respect ducks so, uh, pie into creating negative two one plug in to eat to the books we&#x27;LL square this and squaring this will have to turn to a force I can pull that out or pie integrated negative to the one. And then there was he to the two packs pukes now integrating e to the two bucks Well, I for pie so it stays the same and then we divide by its new power So too or the we divide by the constant in the power for exponent Jules. And then we want to evaluate this from negative to one So that two and the four Khun simplified to just be two pi and we&#x27;ll end up with this is equal to to pi evaluated at will we plug one in which would just be too too. And then we subtract off when we call negative to end which will be e to the negative four And this here will be our Here we go.

Linh Vu · Answer

they were recorded in the Fulham Echo Juda Fine terms into Google and you be function F X square. The act they were given the a echo Judah zero. Uh, no. Sorry. Could you minus do and be equal to the one on the function F acts? Could you do each of the AKs Now? If I can get the bottom, it will echo chewed up by intercourse from Music Ministry to one. Yes, I would have the Jew each of the X Square. They were getting a cogent by governments. 2 to 1. You have a far evil two x d x. Your father constantly camping outside. Ever have the far Bahir untidy review But about your ex? Every coach should be about your ex but the General Limited dividing baijiu on now, a pirate on the minus. 2 to 1. Yeah, the far Jewish a constant complaining outside and get it Could, you know, troop I only and in good with the one inside. I&#x27;m gonna e about ju minus. It would have my honesty. And so we have that a now all of the minus fall And this will be the answer here. Well,

Problem

Solve the given problems by integration. Find the…

Problem 35 Easy Difficulty

Solve the given problems by integration.
Find the volume generated when the first-quadrant area bounded by $y=e^{x}$ and $x=2$ is rotated about the $x$ -axis. [Hint: After setting up the integral, rewrite $\left.\left(e^{x}\right)^{2} \text { as }\left(e^{x}\right)\left(e^{x}\right) .\right]$

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Basic Technical Mathematics with Calculus

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Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Catherine R.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus