sketch-the-region-enclosed-by-the-curves-and-find-its-area-x1-y-x0-y1-ye

Question

Sketch the region enclosed by the curves and find its area.
$$
x=1 / y, x=0, y=1, y=e
$$

Gregory Higby · Accepted Answer

don&#x27;t let ah, the different notation kind of scare you. Um, X equals one over. Why? Still looks like y equals one of her, actually. And think about cross multiplying and dividing still looks the same. S o X equals zero. Is this vertical line. So why access and then y equals one. Is that why it was one? And I y calls it he is up here. I&#x27;m saying we were looking for this area, and it&#x27;s just in terms of why. So you&#x27;re horizontal lines of your bounds. Um, your upper function is one over Why? And then your lower function is is the ex eagle zero. But you don&#x27;t actually have to write that out cause of trying you zero does not change the value of that eso as you&#x27;re looking at this. Um, when you do the anti derivative of one over why, it&#x27;s just natural log of why from one to e um, And then as you plug it in, you get natural of e minus natural log of one. And I like to think everybody in calculus would know natural law gov is equal to one natural log of one is equal to zero. The one minus zero is one his aunt&#x27;s. We want

Sinisa Stura · Answer

Hello. We&#x27;re given three curves. Why cool into the part of a, uh Why? Equal time? Beat the bricks on the hood. Zero we are classed. Find the area, uh, enclosed by these three. So we spiral by bitching this by hand. Okay, we start with of this invidious position B with the work. I think you know why, uh, with Congress through your little function, why equals f of X equal Arabic. We should not have to sketch this scared because, you know, it&#x27;s past the point each of the power of 01 uh, e to the power of one. Okay, so we have these points here, and he&#x27;s around there. Um, are you for the exponential curve passing through one? Some people. Okay, the next curve is, uh why people Local dysfunction. Jurich. Yeah. 00 So the car will pass through this point. You&#x27;re one. X equals when it&#x27;s warm times, even the power when you go through one. He the same point that we have here on. If we, uh, the hey could be a storm, this will be minus. When this one time, he&#x27;s in a pair of mind. He might be with 30 minutes for this would be somewhere here. All right, this crew. Well, look. Smoking record. Look, the area the area we&#x27;re interested in is this area here. All right, berry between the curved and you, uh, in the sketch, we&#x27;ll see that Hey is between you know why? Because intersection point the intersection point between f of X and you here on the why is that third here? So we&#x27;re two reports determine this region here, uh, to sing Syria more plainly. We may use our friendly online freak out later. A dismal don&#x27;t come. And when we enter the ocean, we see the photograph is the eight to the power of it graph is the other function, Excellency, our vic. And this is the why thank you. The area. This is the area? A little. I want a piece of advice to check your answer. I mean, you can use this. You can use business to find out the, uh, answer to this question. What is the area? Uh, in a moment. Okay. The tool that we will use is this the area of a region. Ah, that is bounded. Bye function. It will be which is above whose rap is above the great. You like? Yeah, integral with bones Old are in the interval of it. So the interview We&#x27;re talking about the central here, the four eggs belonging to the intro loser in the world We&#x27;re looking What is area off the region bounded by this function about and from below The area&#x27;s bounded by Jim Yeah, All right. So what we will have our integral bomb will be from throughout the world if you slur. If you start fighting I and in the next free here you will, Steve. And fine. And then you just right at this formula. And you will have, uh, the culturally to give you a result 0.0 1 to 8, which we will use the chick with our results in a minute. So off we go Finally area. We need to find integral 04 off function ethics, which is even the power of it, minus the functions. You vixen. You take care of it. Do you, uh, this integral the interval off the interval into the barrel. So this is where we were having a problem. But this part, though we will need to integrate by parts. What does it mean interviewed by power The formula for integration. Bipartisan. We have, uh, the rule of you be Were you the You go through the grate. Excellent. Eaten Arabic we say to be you, uh, could be float Do you is equal to if you could sink the river Vixens one and the business TV interval, Uh, each of the parents of the year. Then we have the You look better, uh, labeling this group. I like foreign rule. We know her by using this formula here. You time to see who. Six integral off the times the you, which is one times of the and take it. Marler needs to the bird thing because because we&#x27;re working with indeterminate in the rules. Here we go back. So a definitive go a sequel. The integral off You need to the power of it. And here we have Martin, The interval of the second term Here. The second room. We&#x27;ll raise it. Thanks for the battery. Yeah, which is evaluated from 0 to 1. Right. Simplify this oppression in the brackets. Either dive a line before the for interesting things sign. Don&#x27;t forget that error that is made on. We have too early. Thanks. You&#x27;re wolf. Let&#x27;s see, We can effect around each of the paralytic. Really evaluated in from a miracle, right? No, we evaluate, uh, substitute takes. Well, we, uh, ankle e super. Whoa. Martin Now zero feet to the fire of zero. Which multiple zero. Hey, which is each and Sarah. Zero is one. Some 10 Who for? B one. Well, this is our goal. If we go back, just sit back. Remember what I said in this loss that you will get this result. This is equal to the murder. And there we go, because it the thank you.

Kenneth Kobos · Answer

we want to find the area of the enclosed region, uh, closed by the functions y equals one of her ex ah y equals X and y equals 1/4 times X. So doing a quick sketch shows us that the region is this triangle over here. So we do have to figure out what are these three points so that we know what will be our limits of integration. So if we label these points 12 and three, let&#x27;s take a look at one first. So this point is very easy. It&#x27;s easy to see it&#x27;s the origin because this is the intersection of the points of the lines X and 1/4 X solving. For this toeses X equals zero. Let&#x27;s take a look at the second point. This is the intersection of the line X with the function one over X solving for this gives us X squared equals one which gives us exes either plus one or miners one. But since we&#x27;re only considering X greater than zero, this tells us that X is equal to one is the second point. And now if we take a look at the third point, this is going to be the intersection of the line 1/4 X, uh, with the function one over X. So solving for this we get X squared equals four. And then again, since excess positive X is equal to two so we can label these points over here with, um zero X equals zero X equals one and X equals two. So now we want to integrate to figure out the area. So let&#x27;s open up. A new page area is equal to first. We&#x27;re integrating from 0 to 1, so it&#x27;s going to be the top function, minus the bottom function. The top here is X, and the bottom is 1/4 x so x minus 1/4 X on all of this D X plus the next part We&#x27;re integrating from 1 to 2 and we look at the top function here is one over X, and the bottom function is 1/4 X. So we have one over X minus 1/4 X d x. So this is going to give us the area so we just integrate this. We can do this in one piece, because X minus 1/4 X is just 3/4 x. So here we have the into the anti derivative of 3/4 X is three over eight x squared. This is going from 0 to 1 and then plus, here we take the anti derivatives of these two pieces. The 1st 1 is the anti derivative of one over X Is the natural longer them or lawn of X on. Then we have minus one overeat X squared and this is going from 1 to 2. So after we plug in everything So let&#x27;s write this out. This is three over eight minus zero. Plus, here we have a lot of two minus So two squared is 44 overeat is 1/2 and then we minus. Now we plug in one loan of one is zero and then minus one over eight. So we see that things cancel because here we have a positive one over eight, and here we have a three overeat. So that works out to for over eight, which is 1/2 which cancels with this negative half. And then we&#x27;re left with the answer, which is Lung of two

Sky Li · Answer

again. Let&#x27;s draw those two curves first. So say this is our acts My plane in the first Carvey&#x27;s actually, uh, he&#x27;s just a squared X move A shifted to the right by one. So we know, um, a square hotel fax It&#x27;s look like this. So he&#x27;s shifted by once we start from one, and it goes like this. The second is the street line X minus Y equals one. So but why go to zero X equals one? So across this point and x zero by cosa naked across this part our street flying is here. This is our street line. Next, minus y. You close one. Okay. Ah, and then close area is the shaded regions Now, Alice, trading for in the area. Um, you know, to find this outer area, we need to take the integral by the respect, respect to X or respect why it depends on ah how you represent those functions. Say, since I Personally, I don&#x27;t like this square root because if you remember the square root of something, the anti duty off that is it&#x27;s very difficult to remember and looks ugly. So here I&#x27;ll reform the first function. So I take square on both side I&#x27;LL get up Thanks. He closed Tio one plus y square. Hey, And remember over, um our wise quater than zero. The greater the co two zero right, Because the white goes to a square root off something So it always known Active on this since the first function I represented by Why this I also want to represent But why? So I do Here is some I mean why to the other side. So the second function will be activity close to one plus one. Right? Um after this so many in a row with respect. Why? And let&#x27;s find hungry No, a star from zero. And what about this point? The basis our way too. That was so bad off. Why To just say this happens when y plus by square yukos one plus y And we know why too cannot be zero. Which means I lied too. Ecos water because this equation will give us to swoosh is one of them yourself. One and another is zero since white who is not zero. So it has to be one on our boundary with coast from zero to one in the things I the integral. Always red curve minus the left curve or red car abuse of a bus boy. And now our left curve. Here is this one plus life work. Okay, now, Andy. Dear God, this is one half those one canceled. So the anti dirty off. Why is what? Half white square? Minus one cube. My third one like you. Hey, you graduated and in one zero. So Plugin Lycos one. This is what Half minus one third of the cheese. One six in the second term zero. So fix Ivancic will be our answer.

Amrita Bhasin · Answer

okay. We know we&#x27;re only looking at the positive side because we know in this context, we can only have positive area. Okay, so now we over area is from 01 x cubed Jax. And I remember how we integrate increase The exploited by one&#x27;s own. Just four instead of three were dividing by the new explain it, which is four now we&#x27;re plugging in. This is officer just zero, Which means we have 1/4 is our solution.

Gregory Higby · Answer

So I&#x27;m going to give a very lame graph here because an exponential function. Either the ex looks something like this is the ordered pair. 01 by the way, Actually, doesn&#x27;t matter. But, um so this would be eating my ex now e to the X squared grows exponentially faster. I&#x27;m like this each of the two X um just think if you&#x27;re doubling your exponents, it&#x27;s gonna, well, horizontal squeeze if you will, um, and they cross the y axis at the same point. And the reason why I say it doesn&#x27;t matter is then they also tell you that they want one bound to be X equals zero and the other bound to be X equals. So these air vertical lines, um, natural log of to So what we need to do is the integral from zero the lower bound, which he told you to the upper bound of natural, uh, give to of the upper function Eat of the two x minus e to the X DX. Now I would not use u substitution if you want my opinion as I would just think. OK, well, if it&#x27;s e to the two x well, if I did the derivative of each of the two X I would get either of the two x times to cause a chain rule. Well, how do you fix the times? Two is sticking a one half in front and can double check that. This is correct, cause if you do each other to actually get you to the two x Times two, which cuts out which cancels out with this one half. So we&#x27;re good there, minus the drift of Evita, the exes, you the x from zero to natural log of to. So as we plug things in, um, you get one half eat the to natural log of to minus E to the natural log of to minus one half E to the zero minus B 20 And you might say, Well, two times, you&#x27;re still zero. I just wrote that. Um, Now I would do a lot of work just to make sure I&#x27;m doing this. Correct that, Cheech colors change. Okay, I would actually show the work of moving this to as an excellent that your law of logs, and then you could cancel this out. But this is really two squared, which is four times. One half. That&#x27;s how you get to is that first piece and then here. Um, either the natural log cancels out kind of the same way. Is that? And so you just left with two. Now, either the zero equals one. So you&#x27;re ending up with one half minus heated. The zero is one. So, as you&#x27;re looking at this end of two months to a zero minus one half, minus one is negative one half Well, zero plus one half equals one half, which is your answer.

Problem

Sketch the region enclosed by the curves and find…

Problem 14 Easy Difficulty

Sketch the region enclosed by the curves and find its area.
$$
x=1 / y, x=0, y=1, y=e
$$

Answer

$$
A=1
$$

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Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals

Grace H.

Anna Marie V.

Kayleah T.

Caleb E.

Integration Review - Intro

Definite vs Indefinite

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$$A=1$$

Calculus Early Transcendentals

Grace H.

Anna Marie V.

Kayleah T.

Caleb E.

Integration Review - Intro

Definite vs Indefinite

Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

Grace H.

Anna Marie V.

Kayleah T.

Caleb E.

$$
A=1
$$

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals