sketch-the-region-enclosed-by-the-curves-and-find-its-area-yx2-ysqrtx-xfrac14-x1

Question

Sketch the region enclosed by the curves and find its area.
$$
y=x^{2}, y=\sqrt{x}, x=\frac{1}{4}, x=1
$$

Gregory Higby · Accepted Answer

Oh, um, So the important part of this problem is that, you know, the squared of X looks, something like this, and then ah, X squared is the parappa. Look, some doing this, um, this is X squared Ah. Which is important to distinguish which one is the upper function. But then they also tell you that you they only want the area between X equals 1/4 which is just a vertical line to the right and actually goes. One is actually where they intersect each other. Um, so this 1/4 by the way, X equals is a vertical line. So that&#x27;s why your own vertical. So basically all you need to do if you&#x27;re finding the area between, um between all of that is ah the integral from the lower round of 1/4 to the upper bound of one. And you do the upper function, which is square to the X right here is above the other and then minus X squared is the lower function right here. We&#x27;re finding this area. That&#x27;s all you have to do. Um, now, obviously a little bit more work in this, because the anti derivative of this you add one to your ex bone of one half. That&#x27;s what the square root is. So that&#x27;s three halves. And then dividing by that new number is the same as multiplying by the reciprocal of two thirds. Um, and then same thing with this one. You add one to your exponents and you divide by your new exponents and you get one third execute. And that&#x27;s from 1/4 to 1. Um, so this is where the math gets a little bit more daunting because you have to plug in one for each these exponents. But what&#x27;s nice is one toe. Any exponent is just one wonder. 30 still one, and then you have to subtract off. Now, this is where it gets a little bit more complicated. Um, two thirds times 1/4 to the three halves. Power my ass. One third times 1/4 the same being cute. I guess I should. Footprint. This is here as well, anyway. Um, so this is easy. Two thirds minus one third is just one third and then this a little bit more difficult, because this means the square root. Um, with the square one. Any power still one. But the square to forest two cubed is 82 times, two times two, and then this woman I don&#x27;t know if you even know. Maybe one third, um, Times four tens, four tens for 64. Now, I&#x27;m not that great moving forward from here. So I&#x27;m gonna use my calculator to figure out what, two thirds times 1/8 minus one third time&#x27;s one 64th ISS. And as a fraction that tells me that that is 5 60 force and one third minus of that answer as a fraction is 4990 seconds. Well, looked weird, but anyway, that&#x27;s your answer.

Sky Li · Answer

uh, okay. Sketch the region closed by the given curves and finance area. So first, let&#x27;s draw those two curves. This is our XY plane. And, uh, the first function is X equals. So why to the power floor? So it looks like a problem, but it grows much faster. So it opens to the right like this. This will be my ex seacoast y to the power four. Oh, and then the second term here, Um, why he goes to the square root of two minus x. Yeah, this we know. Oh, yeah. Um, it can be represented as if we take the square. This is just a wide square equals two to minus X. We move around things. This is equivalent with X equals two two. Wine is Y square. What? Let&#x27;s score. Yeah, and, uh, be careful here. We also need this indication why is greater or equal to zero? Because you know, the square root my equal to the square root of something, So why cannot be negative? Yeah. Okay. And this is also a problem. He opened to the left on a start at two. And open to the left. Okay. They will be enclosed area will be here. Okay. And here is to Here is General Oh, uh, and as we can see, um, since we reformat the second curve right now, everything is represented by why. So if we want to find an area and take the integral with respect to why, instead of facts and and the boundaries at this point in this point, So if I do know this s by one Yes. S y two how to find them? It&#x27;s just, um Oh, be careful here. The enclosed region is actually now the entire thing, because we could this extra condition here when we reformulate the second curve, why should be no negative. So, um, this enclosed area is only this half of them is here. Here is empty. Okay. Um, therefore, we don&#x27;t have this by one. We only have this fight to on our right. One will be zero. It goes from zero to white too. Okay. Mm hmm. So now we know. Mm. Our boundary by one will be terrible. And why to surely satisfy in a way to is positive. So what? You should satisfy two minus. Why square the second curve and the first curve echoes too wide. The power of four. Okay. Mm. Uh, as we can see, if I plug in by coast to one. This equation hopes so you can say are way too. Must be one. Okay. Or you can do the factoring Then you can solve for this, uh, um, polynomial equation, you&#x27;ll find your roots, and the positive route will be our right to, and it has to be one. Um, So the boundary will go from 0 to 1. Um, seeing inside the integral is the record minus left, Curve like curve. Here is just two minus. Why? Square minus the left. Her left her is, uh, right to the power of four. An anti duty of that is to why mine is one third like cube, minus month. Faith, my faith and inveterate one and zero and receiving Michael 20 this whole thing you go to zero. So we only care when y you go to one plug in white. Go to one. We got two minus one third, minus one faith. So it will be on two minus one third. My nurse on fifth. So this this is, uh, 8/15, right? So it will be 30/15 miners, eight or 15. Oh, So this whole thing will be, um, 24. 22 over 15. Mm, 15. Okay.

Gregory Higby · Answer

Oh, it&#x27;s ah, it&#x27;s problems. Pretty straightforward if you memorize everything. So this is the one over the square root of one minus X squared function. And then why close to is like right here. So the area that we&#x27;re looking for just here. Um, now, if I were doing the problem, I would just do half, because this is an even function. I would just do the integral from zero to, I believe this X coordinate where they intersect. Um, What? You can find it by setting them equal to each other. Because it&#x27;s the equation. Y equals two. I forget to write that down. So you said them equal each other and you square both sides. You have 1/1 minus X squared equals four. Then you could cross multiply, I guess so that before minus four x squared. So subtracting for being negative 33 bores, Um, because you have to divide the four over. So I forgot to mention that so square root that and then when you squared it for you get to, um So my bounds, we&#x27;re gonna be zero to root. 3/2, um, of the upper function being to minus your lower function the the fancy one that they gave you. And don&#x27;t forget, because I almost forgot to hit to write down double that this is will help us find the area. Um ah, yeah. So Ah, this would be a waste of time to rewrite that. I don&#x27;t know. No thinking. Um, So the anti derivative of this, I would be two X and then hopefully you&#x27;ve seen this before. That Andruw of his inverse sign of X. You can just quickly google that if you haven&#x27;t learned it before. Um, and that&#x27;s plugging in from zero to root 3/2. So as we plug this in, you said that to in front. So it&#x27;s double route three over to you can see things will start to cancel minus inverse sign over 3/2 and then minus zero times to still zero. Um, and then they turned this into a plus because it&#x27;s subtracting the negative inverse sign of zero. Um, but if you look on the unit circle, where to sign equal Route 3/2 will signs the light coordinate. That&#x27;s at this order, payer. One half brewed, 3/2. So we want that one. So if you&#x27;re not familiar with the unit circle, that&#x27;s pi or three by over three. And then Warda signing zero. That&#x27;s the Y Coordinate right here in the radiance. Right. There is zero. So this would happen to be zero as well. But you don&#x27;t Never, never heard Siddle check. Um, you might just leave your answer like that. Some teachers prefer it that way. Or distribute that to in because of my right, I only did half on the area. I went from zero to root, 3/2 sided double over here, um, to buy or three. But I like this answer better.

Fuzail Shakir · Answer

All right. So for this question, we have X squared and negative X squared plus four x You have X squared is equal to Native X squared plus or acts we have to find the intersection. Points just becomes X squared plus two x minus four X is equal to zero plus x squared, so this is equal to two X squared minus four. X is equal to zero with a factor of two. X to get X minus. Two is equal to zero, so we have actual equal to two on X is equal to zero. So therefore we can rewrite the integral as the integral from 2 4 0 2 2 0 2 0 0 to 2 of the ex queer minus negative X squared plus four X is equal to be integral from 4 0 to 2 of x two X squared minus four x This is to third X cubed minus two X squared months. We evaluate this from zero to and that that this is equal to eight over three

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.

Problem

Sketch the region enclosed by the curves and find…

Problem 7 Medium Difficulty

Sketch the region enclosed by the curves and find its area.
$$
y=x^{2}, y=\sqrt{x}, x=\frac{1}{4}, x=1
$$

Answer

$$
\frac{49}{192}
$$

Related Courses

Calculus Early Transcendentals

Related Topics

Discussion

Top Calculus 2 / BC Educators

Caleb E.

Kristen K.

Michael J.

Joseph L.

Calculus 2 / BC Courses

Integration Review - Intro

Definite vs Indefinite

Recommended Videos

Watch More Solved Questions in Chapter 6

Video Transcript

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals

Caleb E.

Kristen K.

Michael J.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$$\frac{49}{192}$$

Calculus Early Transcendentals

Caleb E.

Kristen K.

Michael J.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

Caleb E.

Kristen K.

Michael J.

Joseph L.

$$
\frac{49}{192}
$$

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals