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Integral du over under betting

Cross out that row, that column, 1 times 1 minus 0. So it's going to be plus k. So if we want the magnitude of this, this whole thing right over here is just going to be the square root-- I'm just taking the magnitude of this part right over here, the actual cross product.

It's going to be negative 1 squared, which is just 1, plus negative 2v squared, which is 4v squared, plus 1 squared, which is just 1. So this whole thing is going to evaluate to-- we have 2 plus 2v squared du dv. Or actually, let me-- I almost made a mistake. That would have been a disaster. And if we want, maybe it'll help us a little bit if we factor out a 2 right over here. This is the same thing as 2 times 1 plus 2v squared du dv. If you factor out the 2, you get this is equal to the square root of 2 times the square root of 1 plus 2v squared du dv.

And I now think we are ready to evaluate the surface integral. So let's do it. All right. So let me just write this thing down here. So I'm going to write everything that has to do with v, I'm going to write in purple. So I'm just going to write the ds part right over here. It is the square root of 2 times the square root of 1 plus 2v squared. And then we're going to have du, which I'll write in green, du, and then dv. So this is just the ds part. This is just the ds. Now, we have the y right over here.

All right, so I'll write that in purple. So y is just equal to v. That is y-- let me make it very clear-- and all of this is ds. And now I can write the bounds in terms of u and v. And so the u part, u is the same thing as x. It goes between 0 and 1. And then v, v is the same thing as y, and y goes between, or v goes between, 0 and 2. And I now think we're ready to evaluate.

And the u and v variables are not so mixed up, so we can actually separate out these two integrals, make this a product of two single integrals. The first thing, if we look at it with respect to du, all this stuff in purple is just a constant with respect to u, so we can take it out of the du integral.

We can take all this purple stuff out of this du integral. And so this double integral simplifies to the integral from 0 to 2 of-- I'll write it as the square root of 2v times the square root of 1 plus 2v squared. So I factored out all of this stuff. And then you have times the integral from 0 to 1 du, and then times d, and then you have dv.

And now, if this was really complicated, I could say, OK, this is just going to be a function of u. It's a constant with respect to v. And you could factor this whole thing out and separate the integrals. But this is even easier. This integral is just going to evaluate to 1. So this whole thing just evaluates to 1.

And so we've simplified this into one single integral. So this simplifies to-- and I could even take the square root of 2 out front-- the square root of 2 times the integral from v is equal to 0 to v is equal to 2 of v times the square root of 1 plus 2v squared dv. And so we really are in the home stretch of evaluating this surface integral. And here, this is basic. This is actually a little bit of u substitution that we can do in our head.

If you have a function, or kind of this embedded function right over here, 1 plus 2v squared, what's the derivative of 1 plus 2v squared? Well, it would be 4v. And we have something almost 4v here. We can make this 4v by multiplying it by 4 here, and then dividing it by 4 out here. This doesn't change the value of the integral.

And so now this part right over here, it's pretty straightforward to take the antiderivative. The antiderivative of this is going to be-- we have this embedded function's derivative right over here, so we can kind of just treat it like an x or a v. So this is the antiderivative of that. And then, of course, you still have all of this stuff out front-- square root of 2 over 4 And we are going to evaluate this from 0 to 2. We don't have to worry about that at 2 and 0.

And actually, this will cancel out. That becomes a 1. That becomes a 2. And now if you evaluate this at 2, you have 2v squared. So it's going to be equal to And then minus this thing evaluated at 0. Well, this evaluated at 0 is just going to be 1. Bronshtein K. Primitive function. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, log in to check access. In certain cases the domain of definition of a primitive function F x is wider than that of the given function f x. If the function f x is defined in a connected domain except for isolated points of discontinuity x 1 , x 2 ,…, x n , then the domain of definition of its primitive function may contain these points see p. Google Scholar. See footnote on p. For tables of elliptic integrals see p.

For the graphical integration construction of the graph of a primitive function from that of the given function see pp. The numbers A , B ,…, L can also be obtained by the method of undetermined coefficients see p. The polynomials P 1 x and P 2 x can be found easily, if the factors of P x are known, i.

In cases, when the integrals A are expressible in terms of elementary functions, they are called pseudo elliptic integrals. The symbol R denotes a rational function of the expressions to which it refers. Integrals of functions involving sin x and cos x together with the hyperbolic functions and the function e ax are given on pp.

In this case, Li x should be understood to be the principal value of the improper integral see p. The integral logarithm is related to the integral exponential function see p. The concept of the definite integral can also be generalized for functions defined in an arbitrary connected interval an open interval, an interval open on one side, a half-axis or the whole number axis or in a connected interval except a finite number of separate points.

The integrals considered in this more general sense belong to the improper integrals see p. The definite integral exists also for any bounded function with a finite number of points of discontinuity in the interval [ a , b ]. A function for which the definite integral exists in the given interval is called integrable in this interval. In order to avoid misunderstanding, the variable of integration has been denoted by t see p.

In a general case, when the direction of the force does not coincide with the direction of the motion, the work can be calculated as a line integral see p. For the Gamma function see p.

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He had hoped to establish some closer communication with the Jewelbright. Integrals of Trigonometric Functions: When it comes to finding the integrals of trigonometric functions, we often need to use a number of different trigonometric identities and girl farts in mouth I stopped, trembling, and stared at him for a few long, horrible seconds. He just opened his arms, and I fell into them, sobbing my heart out as he stroked my hair.

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She looked like the spirit of an elusive early English summer, when the blossom is still clean and the air dappled with cool and shifting light. I am compelled to pronounce it tobacco smoke. But now that I look again, you are not entirely wrong: there is a considerable admixture of jute, moss, and I think tallow.

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If he sent her back to Tortoise Lane would she tell Mrs. Mapes what he had asked her, and where she had taken him. He did not hold much hope that she would hold out against Mrs. Nellie lived in a cocoon of hunger and fear. Tortoise Lane was her home-probably all she knew. Voskavich left then and Rozhdestvenskiy stood amid the bodies, by the sealed doors, studying the faces.

They had died for something important-what was it. Pulling it open toward him, he shone his light inside. He felt like Carter at the discovery of Tutankhamen. No golden idols were here, but file cabinets, unopened, unlike the ones in other parts of the complex. There was no pile of charred papers and microfilm rolls in the center of the floor. There was community responsibility for health, welfare, and education.

What of those who were disgruntled over their assignments. What of those who objected to the cooking or to living in such tight quarters. What of the clash of personalities. The captain did not turn around as the journalist entered. Her thin, bony face was expressionless. She looked around the deserted bridge. I may be uneasy about it, but I intend to have the job done right.

A Tudor named Henry the Tenth on the English throne, and a North American federation of Inca chiefdoms with not much technology beyond good roads and the telegraph. And then I remembered another thing, which caused me very heartily to curse myself. It came from the Marat tableau all the time.

Now all that money meant nothing. He heard shots behind him and felt a slug rip past his ear. It seemed to pull him along and he ran faster. Another bullet gouged splinters off a tree in front of him. Augusta had known her too long to be deceived by the arrogant stance, the squared shoulders under the thick satin. Good heavens-she was still so much of a child, as feckless as a summer day. She saw what she wanted and reached for it.

Where on earth had she come by such abandon. It was certainly not her father. But it was not his place to ask questions. Did that come from the shrinking of the planet-born who had never been in space, or was it a cautionary impulse triggered by something else. There were a number of attendants around but none of them showed the characteristic features of the Asborgan-born. These were mainly off-worlders and some were truly alien.

In this very mixed group he might not have attracted the general eye, for he was wearing the livery of a high lowland house and accompanying a young Highblood. Surface integral example part 2. Surface integral example part 3: The home stretch.

Surface integral ex2 part 1. Surface integral ex2 part 2. Surface integral ex3 part 1. Surface integral ex3 part 2. Surface integral ex3 part 3. Surface integral ex3 part 4. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript Now that we've set up the parametrization, let's try to evaluate the integral. And the next thing we'll do is essentially try to express ds right over here in terms of du and dv. And we've seen this before.

So first, let's take the cross product, and we'll do that with a 3 by 3 matrix. I'll do that right over here. We set up, and I'll just kind of fill in what r sub u and r sub v is in the actual determinant right over here. So first we have our components, i, j, and k.

And now first, let's think about what r sub u is, the partial of r with respect to u. Well, its i component is going to be 1. The partial of u with respect to u is just 1, so its i component is 1. Its j component is going to be 0, partial of v with respect to u is 0. This should be parentheses around here. Partial of this with respect to u is just going to be 1 again. The partial of v squared with respect to u is just 0. So this is just 1 again. And then r sub v, the partial of r with respect to v, the i component is going to be 0, j component is going to be 1, and the partial of u plus v squared with respect to v is going to be 2v.

So it's a pretty straightforward determinant, so let's try to evaluate it. So the i component, it's going to be i. So we cross out this column, this row. It's going to be 0 times 2v minus 1 times 1. So essentially, it's just going to be negative 1 times i. So we're going to have negative i, so this is equal to negative i. And then the j component-- and we're going to have to put a negative out front, because, remember, we do that checkerboard pattern.

So cross out that row, that column. Let me make sure I got that. So it's just going to be 2v. But since it was the j component, which is going to be negative, it's going to be negative 2vj. Let me make sure I did that right. That column, that row, 1 times 2v is 2v minus 0 is 2v. The checkerboard pattern, you'd have a negative j. So you have negative 2vj. And then we have find the k component. Cross out that row, that column, 1 times 1 minus 0.

So it's going to be plus k. So if we want the magnitude of this, this whole thing right over here is just going to be the square root-- I'm just taking the magnitude of this part right over here, the actual cross product. It's going to be negative 1 squared, which is just 1, plus negative 2v squared, which is 4v squared, plus 1 squared, which is just 1.

So this whole thing is going to evaluate to-- we have 2 plus 2v squared du dv. Or actually, let me-- I almost made a mistake. That would have been a disaster. And if we want, maybe it'll help us a little bit if we factor out a 2 right over here. This is the same thing as 2 times 1 plus 2v squared du dv. If you factor out the 2, you get this is equal to the square root of 2 times the square root of 1 plus 2v squared du dv. And I now think we are ready to evaluate the surface integral.

So let's do it. All right. So let me just write this thing down here. So I'm going to write everything that has to do with v, I'm going to write in purple. So I'm just going to write the ds part right over here. It is the square root of 2 times the square root of 1 plus 2v squared. And then we're going to have du, which I'll write in green, du, and then dv. So this is just the ds part. This is just the ds.

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