in-example-512-we-showed-that-displaystyle-int1_0-x2-dx-frac13-use-this-fact-and-the-properties-of-i

Name: in example 512 we showed that displaystyle int1_0 x2 dx frac13 use this fact and the properties of i
Uploaded: 2019-12-01
Duration: 25 s
Channel: Numerade
Description: In Example 5.1.2 we showed that $ \displaystyle \int^1_0 x^2 \, dx = \frac{1}{3} $. Use this fact and the properties of integrals to evaluate $ \displaystyle \int^1_0 (5 - 6x^2) \, dx $.

Question

In Example 5.1.2 we showed that $ \displaystyle \int^1_0 x^2 \, dx = \frac{1}{3} $. Use this fact and the properties of integrals to evaluate $ \displaystyle \int^1_0 (5 - 6x^2) \, dx $.

Amrita Bhasin · Accepted Answer

Okay. We know the original into girls. The order 15 minutes to click Squared DX could be broken up into five D X minus six x squared DX like this. Therefore we cannot calculate. This is a quick 126 simply five months two, which is three

Katelyn Chen · Answer

you&#x27;re told a girl from 0 to 1 of x 40 x is equal to 1/3 were asked to determine integral loser one of five months. Six X cords, Yes. So start off with that. We can use. We can separate are integral into the uncle phone. Soto want five minus integral. Observe the want the six x squared DX. We can simplify this that I am. I bring the constant sex out. Get 10 above X squared d X We noted integral from 0 to 1 of five has a wreck, Taylor with of five. And that five just straight line. So this would just be equal to one time socks. That&#x27;s just equal to five my six times. And we know that this integral, we&#x27;re told, is equal to 1/3 which is equal to five minus. To which he gets us three as her answer

Robert Daugherty · Answer

Yeah problem. 44. They want us to evaluate using properties of integral roles. The integral 1 to 3 to eat of the X men is one given that in a previous exercise. A few pages before this they work through an example and found this result. So properties of integral is here. This is going to be the integral from 1 to 3 of two. E to the X D x minus. The integral from 1 to 3 D X. That is to times the integral from 1 to 3. Either the X D x minus. The integral from 1 to 3 D X. Now the result that you see here, I already know that value so I can substitute that in. So this is going to be yeah, two. Mhm E cubed minus E minus. And this is just going to be three minus one. So when you do that integral is going to be So 3 -1. The reason you see this is, what does that function look like? This is like, well, one if y is equal to one from 1 to 3. Yeah, so here&#x27;s one, here&#x27;s one, here&#x27;s three. What is the area of this rectangle? That is a one by three rectangle? Excuse me? one x 1 x two Rectangle. So it turns out this is just simply two E cubed minus E -2. Or if you factor to out of everything, EQ -E -1 is the final answer there.

Nathalie Luna · Answer

soon in this problem, we are asked to evaluate this integral. And so the first thing that we notice spotted it&#x27;s that it&#x27;s an integral off the power eggs and were derived integrating with respect to X. So we the first thing that we&#x27;re going to recall this tower pour into girl. So for integrating X lay still power and where in has to be different from one. Uh, this tentacle will be ex raced and plus one and we divide bye and plus one to this is a indefinite And the growth we are, eh? A constant in this case, we don&#x27;t really need it. But so in our case and ISS minus sixty five. So first thing that we want to do to integrate this pendant in plus one. So since this is a fraction of going to do it separately, yeah, and plus one isthe over six from seven hundred six over five plus one. But instead of range one since we want to be able to add a two two numbers were one of them is a fraction they have to have the same the number denominator. So we want something with sign on the bottom Since we&#x27;re talking about one, we want same number asked in a greater and greater. So instead of writing one working to write five five. Still, this is equal to five days right here and now top arguably. Haps my six plus fine. And that is people too. Minus one over. So this isthe class. Fine. Which is what we&#x27;re gonna do. Okay, uh, look. And here and here are integrated. So and this first up, we don&#x27;t need this practice. We&#x27;re already integrating, and we have ex place to n plus one, which we&#x27;ve figured out this minus one over pi Delighted by the scene number Mendes one over. And this expression, it&#x27;s going to be when x equal one and up to thirty two. So before, without practice, huh? Function in thes and points of the integral. I&#x27;m home to reeling to this so that it&#x27;s easier to work with. So we have, uh, this minus saying we&#x27;ll keep it here. We could put it instead over here instead of here. And this five since it&#x27;s in the numerator after the narrator, we can bring it back here to bring it up floor here. All right, so we can rewrite it like minus five eggs. Fruit base, too. Mice one. Over and again. We don&#x27;t lose this X equal to one, two, thirty two. All right, so now, before we do this next part, which issue seemed the fundamental Purim of calculus where we evaluate this when ex house and one x thirty two minus the same expression on except one we are going to I remember some of that fact. So here we have. When we raise one to any power, get one and thirty two, we can write that us to the park soon. I am going to first one X equals to this first part is when X equals thirty two, which we&#x27;re going to raid us two to the fifth power and then we&#x27;re going to use the same expression when X equals one. All right, so let&#x27;s move this downtown. So we have Manus fire. That&#x27;s a constant Tim Specs, just two to the five. And this text is going to be raised to one. Men&#x27;s went over and bring to this is the expression on X equals thirty two and three Subtract the same expression on X equals y. So the constant does not change eggs. This one and the power. All right, so you know, it&#x27;s all constants. We can no work on getting this a little bit. One similar simplifying this right? So we have. And this power, it&#x27;s five times who won. Minus whatever fight the chain. Since this is denominator, the numerator and denominator thes cancel out. So we&#x27;re left with two races to the minus one, which is people to one half. And in this case, since we have one race to some power, this is going to be equal to just so this simplifies too. Minus five times one half between kin, Right? Like this. And this place no, outside this minus and minus turned into positive. And this is five times one. The chains. It&#x27;s just five. Now again, tow up to numbers we have. This is a fraction and a hostin a mayor too. So we&#x27;re going to put it to here. And since we do it on the bottom of this number, we&#x27;re goingto have to do it the same. No matter. So think of this five as five over one. That&#x27;s what it is. And then we&#x27;re just multiplying by to both the numerator generator. Right? So now we can actually at these numbers till you have minus O on the bottom. We have to. That does not change. And we have minutes five plus times two. And that&#x27;s equal to five. Number two seven. That isthe our answer when we evaluate this tibia.

Ian Maurer · Answer

we&#x27;re gonna get a girl from 1 to 32. Have x the negative 6/5 d X. So when we do an anti derivative, we add one to our exponents. So I&#x27;ll have X. So I&#x27;m adding 5/5. So that&#x27;s why it gets the negative 1/5 and you divide your new exponents. Negative 1/5 when you evaluate that from 1 to 32 32 is kind of a suspicious number. I&#x27;ll bring that make up here 32 to the negative one. Oh, come on. One fifth divided by negative 1/5. That&#x27;s F A B minus F of a one to the negative, 1/5 over. Negative 1/5. That&#x27;s that way. Um, remember that the fifth power or the 1/5 power is the same as doing like the fifth root. And so I can actually think of this as as asking me to do the fifth root of 32 to the negative one over negative 1/5 over here. It doesn&#x27;t matter what the power is. For one, it&#x27;s just one over negative 1/5. The fifth through 32 is two. So if two to the negative one in the over negative 1/5 minus. There&#x27;s two negatives over years. We&#x27;ll make that into a plus. And one divided by 1/5 is five. If you bring that down here, we actually just have Ah, two to the negative. One is 1/2 times, uh, negative 5/1 plus five. So we end up with negative five house plus five, which is positive five house.

Amy Jiang · Answer

Okay, so, no. So we have hacker box sign interests, and we know that the derivative of that is one over this term here, so we can use yourself. We&#x27;ll let u equal. High gridlock sinjin verse affects. So do you. That&#x27;s equal to one over square root of X squared plus one. Yeah, So I could rewrite every in a girl. Uh um, you do you? And that&#x27;s difficult to you squared over two plus e. But you write that in terms of you. So that&#x27;s singe in verse of X over two. And then this is squared and we&#x27;ll evaluate that. This has over four on 5/12. So we get sine inverse of 3/4 squared over two minus. Sign in verse of 5/12. Word over, Chip.

Amrita Bhasin · Answer

use this result. Who&#x27;s to evaluate the other? Hear you, sir, We use ah ah, linear property of a crow Writers in the cross thiss this integral is your integral constant function. We can compute this by considering the area of a rectangle with pace because we want. So that&#x27;s interval from zero to one and hide his value of the function, which is what So this gives you are five minus six times this one, this one eyes given by the problem. So it&#x27;s one over three. So this people&#x27;s five minus two, which is three.

Amrita Bhasin · Answer

given that we are told the end and grow from 0 to 1 of expert is one third. We could go ahead and split up the integral that were given, which is just a little bit more complex here. So we have the integral from 0 to 1 of five the X and then minus. We could push that sex out front and then keep the integral from zero the one of X squared the X. But like we said, we know that all of this is equal to one third. Therefore, all of this here is equal to one third, so that would be six 1 30 and then the other, integral is just going to evaluate five. Um, at this point, we don&#x27;t to anti differentiation yet I don&#x27;t believe, but it&#x27;s anti derivative is five x from 0 to 1, and that ends up evaluating for just five. So then we end up doing five minus six times the third six times. A third, of course, is just too so five minus two, and therefore this gives us a final solution of three

Amrita Bhasin · Answer

Okay in this problem we already know that the definite integral from 0 to 1 of x squared comes out to equal one third. Uh So we want to use that fact and properties of integral to evaluate the definite integral from 0 to 1 of to function five minus six X squared dx. Uh Well there&#x27;s a couple of properties of integral that we&#x27;re going to use. Uh The integral of a difference is the difference of the integral. Uh So this is going to equal the integral from 0 to 1 of five DX. Subtract The integral from 0 to 1 of ah six x squared dx. So let&#x27;s rewrite integral from 0 to 1 Of five GX. Subtract Now the other property of integral is that we&#x27;re about to use is the integral of a constant times of function is equal to the constant times the integral to function in other words, the integral, six x squared is really six times the integral of x squared. So we&#x27;re going to take the six out of the integral sign. Um So six is now going to times the integral from 01 of x squared dx. Yeah, okay, when you integrate a constant but the definite integral of a constant is simply that constant Times uh the interval of integration we&#x27;re integrating between zero and one. So the length of that integral is 1 0 or just one. So at the end of roll of five D x from 0 to 1 is just five Time to lengthen the interval 1 0 which of course will be five times one. And then we are subtracting six times the integral from 0 to 1 of x squared dx. But this was given to us in the original problem. Uh The answer is one third, so six is really just times in a value of one third, So five times 1 0, 1 0 is one. So 5 times one is 5. Subtract six times one third is to, Final answer is three.

Problem

Use the properties of integrals and the result of…

Problem 43 Easy Difficulty

In Example 5.1.2 we showed that $ \displaystyle \int^1_0 x^2 \, dx = \frac{1}{3} $. Use this fact and the properties of integrals to evaluate $ \displaystyle \int^1_0 (5 - 6x^2) \, dx $.

Answer

3

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