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PROFESSOR: So we go back to the integral. We think of k. We'll write it as k naught plus k tilde. And then we have psi of x0 equal 1 over square root of 2pi e to the ik naught x-- that part goes out-- integral dk tilde phi of k naught plus k tilde e to the ik tilde x dk. OK. So we're doing this integral. And now we're focusing on the integration near k naught, where the contribution is large. So we write k as k naught plus a little fluctuation. dk will be dk tilde. Wherever you see a k, you must put k naught plus k tilde. And that's it. And why do we have to worry? Well, we basically have now this peak over here, k naught. And we're going to be integrating k tilde, which is the fluctuation, all over the width of this profile. So the relevant region of integration for k tilde is the range from delta k over 2 to minus delta k over 2. So maybe I'll make this picture a little bigger. Here is k naught. And here we're going to be going and integrate ...

PROFESSOR: Welcome back to recitation. In this video we're going to do another solid of revolution problem. So what I'd like to do in this problem is to find the volume of the solid generated by rotating the region bounded by the following curves: y equals 0, x equal 4, and y equals square root of x, around the line x equals 6. And you can choose your favorite method to do this. What I would like first, when you're doing this kind of problem, is get a rough sketch of the region. So you have some picture of what's actually going to happen. You don't necessarily need a three-dimensional picture. But at least have the two-dimensional region and understand the where the rotation line is, with respect to that region. So I will give you a little bit of time to work on that problem. And when I come back I'll show you how I do it. OK, welcome back. So again, what we're doing in this video is we're going to be looking, finding the volume of a solid ...

JOEL LEWIS: Hi. Welcome back to recitation. In lecture, you've been learning about triple integration. And I have a problem here for you on computing a volume of a region using a triple integral. So let's look at this. So I have a volume and I'm describing it to you; it's the volume inside the paraboloid z equals x squared plus y squared and bounded by the plane z equals 2y. So I've drawn a little picture here for you. So this is the paraboloid here. And we're just taking a plane cut of it. And so this is going to slice off some chunk of that paraboloid, and what I want to know is, what's the volume of that piece that gets cut off by that plane there? So below the plane and above the paraboloid. So, why don't you pause the video, take some time, work out this problem, come back, and we can work on it together. I hope you had some luck with this problem. I think it's a bit of a tricky one, so let's start to work through it together. ...

JOEL LEWIS: Hi. Welcome back to recitation. You've been doing some work on trig integration. I have a nice example here of a problem that requires trig integration in order to solve. So what I'd like you to do is to compute the volume of the solid that you get when you take one hump of the curve y equals sine a*x and you revolve it around the x-axis. So you take the curve between two consecutive roots, and then you, you know, revolve that around the x-axis and that gives you some, I don't know, vaguely football-shaped thing. And so then the question is, what's the volume of that solid? So why don't you pause the video, take a little while to work that out, come back, and we can work it out together. Welcome back. In order to solve this problem, we just are going to apply our usual methods for computing a volume of a solid of rotation. So in order to do that, remember that one of the things you need is you need to know the region over which you're i...

JOEL LEWIS: Hi, welcome back to recitation. In lecture, Professor Miller did a bunch of examples of integrals involving trigonometric functions. So I thought I would give you a couple more, well I guess three more examples, some of them are a little different flavor than the ones he did, but some nice examples of some integrals you can compute now. One thing you might need for them is you're going to have to remember some of your trig identities. Just like you had to remember some of them last time. So in particular, one identity that you might need today that you didn't need in lecture, was the angle sum identity for cosine. So let me just remind you what that is, it says the cosine of a plus b is equal to cosine a cosine b minus sine a sine b. So you're going to need that formula to compute one of these three integrals. So why don't you pause the video, take some time to work these out. Come back you can check your answers against my work. Hopefully you ...

PROFESSOR: Welcome back to recitation. In this video what I'd like us to do is practice Taylor series. So I want us to write the Taylor series for the following function, f of x equals 3 x cubed plus 4 x squared minus 2x plus 1. So why don't you pause the video, take some time to work on that, and then I'll come back and show you what I get. All right, welcome back. Well, we want to find the Taylor series for this polynomial f of x equals 3 x cubed plus 4 x squared minus 2x plus 1. So what I'm going to do is I'm just going to write down Taylor's-- or the expression we have for the sum, for the Taylor series in general and then I'm going to start computing what I need and I'm going to see what I get. So what do I need to remember? Well let's remind ourselves what the formula is. We should get f of x is equal to the sum from n equals 0 to infinity of the nth derivative of f at 0 over n factorial times x to the n. So that's what we wan...

Hi. Welcome back to recitation. We've been talking about Taylor series for a number of functions and rules by which you can compute Taylor series. I have here an example that I don't think we did in lecture. So this is the function f of x equals secant of x. Now, unlike some of the other ones you've seen, there's not a really simple formula for the whole Taylor series of secant x. So what I'd like you to do is not to find, you know, a formula for the general term, but rather, just to use some of the tools that we've learned to compute the first few terms of the Taylor series for f of x equals secant x. Say, up through the x to the fourth term, if you wanted, or even a little further if you were feeling ambitious. So why don't you pause the video, have a go at that, come back, and we can do it together. So welcome back. I asked you to compute the first few terms of a Taylor series. One thing you can always do in this case, is you can go and you ...

JOEL LEWIS: Hi. Welcome back to recitation. In lecture, you've been learning about using gradients to compute tangent planes to surfaces. So I have an example of a practice problem here for you. So what I'd like you to do in part a is to use gradients to find the tangent plane to the surface z equals x cubed plus 3x y squared at the point (1, 2, 13). And in part b, I'd like you to do something similar, which is to use gradients to find the tangent line to the curve x cubed plus 2xy plus y squared equals 9 at the point (1, 2). So why don't you pause the video, have a couple goes at those. Come back and we can work on them together. So hopefully, you had some good luck working on these problems. Let's just take a look at them. So for part a, you're given a function in the sort of usual form that we use to graph it, which is you're given z equals a function of x and y. But in order to apply this gradient method, what we really want is we want to l...

CHRISTINE BREINER: Welcome back to recitation. In this video, I'd like us to work on the following problem that has to do with tangent planes and approximations. So we know that a rectangle-- we'll say a rectangle has sides x and y, and we know that to find the area of the rectangle then, we just take x times y, and I would like us to approximate the area for x equal to 2.1 and y equal to 2.8. And obviously, with this type of equation, it's not hard to just compute this, but I'd like us to use the tangent plane approximation to determine the value, and then we'll compare it to the actual value, just to give us an idea of how we can use the tangent plane to approximate things. And obviously, I'd like to do this near x equal 2 and y equal 3. So when you're doing your tangent plane approximation, do the approximation at x equal 2 and y equal 3. And then when you're done with that, I would like you to answer this question. So near x equal 2 and...

JOEL LEWIS: Hi. Welcome back to recitation. You've been learning in lecture about matrices and their various applications, and one of them is to solving systems of linear equations. So I have here a system of three linear equations for you. 2x plus c*z equals 4, x minus y plus 2z equals pi, and x minus 2y plus 2z equals minus 12. So what I'd like you to do is the following. Find the value of c-- or all values of c-- for which, first of all, there's a unique solution to this system. Second of all, for which the corresponding homogeneous system has a unique solution. So remember that the corresponding homogeneous system is the system where you just replace these constants on the right by 0. So it's a very similar-looking system. The left-hand sides are all the same, but the right-hand sides are replaced with 0. So you want to find the value of c for which this system has a unique solution, the value of c for which the corresponding homogeneous system has a u...

Hi. Welcome back to recitation. In class, one of the things you've talked about recently was computing surface areas of solids of rotation. So I have a nice problem relating to that here. So the circle with center (R, 0) and radius little r-- so this is center big R, 0, and radius little r, which is less than big R-- is rotated around the y-axis. And the question is, what's the surface area of the resulting solid? So we have here the circle. Its center is at the point big R, 0, and its radius is little r, so this is the equation of that circle. And we're going to rotate this circle around this axis. So we're going to spin it around. And what you're going to get is a donut, or what mathematicians call a torus. So here's a little schematic of it here, with one dotted little cross section corresponding to this circle. So the question is, what is the surface area of this torus? So why don't you pause the video, take a few minutes to work this probl...

PROFESSOR: Welcome back to recitation. In this video, what I'd like us to do is, do a little bit of practice with sigma notation. So this will be just a few short problems to make sure that you're comfortable with what all the pieces in the sigma notation actually do. We're going to start with two problems here. And the first one is going to be a fill-in-the-blanks type of problem. And the object is, I've given you a sum on the left-hand side, and then I've given you two other sums, but I've left in each place two blanks, and I've filled in the rest. You have enough information to fill in the two blanks. So what I'd like you to do in this problem is fill in the two blanks so that the sums are equal. And the object is obviously is to do this without writing out all the terms and adding up and then going backwards. So you really want to try and understand what each part of the sigma notation does. The second problem I'd like you to do is ...