 Welcome everyone. If you're watching this video, you're probably trying to prepare for the first exam in Math 1220, Calculus II at Southern Utah University. As usual, I'm your professor, Dr. Andrew Misseline. I'm going to help you get ready for this exam. I should mention that this is the first exam for our class. So there's a few things I want to make mention in this video that I probably won't make mention in future videos. But what you see in front of you right now is a copy of the practice exam for exam one. You should hopefully be able to find this on Canvas, our learning management system for the class. If you have not already downloaded or found the practice exam, you probably are going to want to do that. Pause this video and find it. You also want to look for the exam syllabus, which hopefully you can find on Canvas in the exact same location. Probably you can find it because you have found this video. But nonetheless, pause the video right now and find it if you need to. The exam for our class does represent a big chunk of our final grade, and so we do want to take it seriously. In this video, I'm not going to talk about semester-specific things. Like things like dates, times, locations of exams, these can change from semester to semester. Please consult the exam syllabus for the semester-specific details like that. For this exam though, what I can't say is I want to talk about the structure of the exam, and I also want to focus on the topics that will be in this exam because these are things that don't change from semester to semester. Now, what you see right in front of you right now is the cover page for this exam. It does include some instructions that you should be familiar with, and like I said, as this is the first exam, there's a few things I want to make mention. First, our exams in this class will come with three different types of questions. The first sections will be called the multiple choice section. The multiple choice questions, which we'll see some of those in just a second, but if the name suggests, you'll be given a prompt, a question, and then you have multiple choices that you could choose from. Among those choices, there's one and only one correct answer. To get full points for that question, you should select the single correct multiple choice option. You would just select it on the page. You don't need a scan drawn or anything like that. And then you don't select anything else. You don't select more than one answer. If you select a wrong answer or you don't select any answers or you select multiple answers or something like that, you wouldn't get any points on this one. This multiple choice question, you get all the points for selecting the one single answer. You don't need to show any work on this, on these type of questions. You won't be graded on your work whatsoever. You'll be graded only upon your selection. So these are questions which are all right or all wrong. Related to those, and so I should mention these multiple choice questions are generally on the easier side since you can't get any partial credit. They should be mostly just one, two step type problems, fairly routine, nothing too elaborate. We'll see some examples of those in just a second. The second type of question will be the short response questions. The short response questions are questions where, well, as the name suggests, they're meant to be short. So in terms of length and difficulty, they are very much similar to multiple choice questions, although they generally might be a little bit harder than multiple choices. Some of them are just like multiple choice questions, but the format of the question doesn't lead itself to a multiple choice. A short response is a better way of asking the question. With a short response question, to get full points, all you have to do is write the correct answer. There is a line provided, a blank line, which ideally I would want you to write the answer on that line, but that's how you get credit. So it might be something like calculate an anti-derivative in which case you didn't write the anti-derivative. There is space provided on the page so that if you do want to show some work or work out the problem, you can do that on the test packet itself, but to get any credit on a short response question, you do not need to show any work. Now, because if your answer is, if your answer you wrote is correct, you get all the points from that. Now, if your answer is incorrect, you can get partial credit because the answer you wrote might be partially correct. Like if you're writing an integral, you have the bounds incorrect, but you have the function correct. You can still get some partial credit there. And also any work you show might earn you some partial credit as well. It's not necessary for full credit, but it might be a good idea to include that. But like I said, it's not necessary. The last section of the exam will be what we call the free response. Maybe the name long response would be better, but that's a little bit more intimidating. So we'll call it free response. You can write anything you want. These are questions for which you must show all supporting work in order to get full credit. If you write the correct answer and show no work, you might get no points on that problem because this one is intended to be longer problems, but I want to see what you wrote to get full credit. There'll be a large blank spot on the page so that you can fill in all the work that you have in there. Now, if there does come a time where you want some scratch paper, you can get some scratch paper. Be aware that the scratch paper will be discarded at the end of the exam and will not be graded. So make sure everything you want graded is inside the test packet for these types of things. The number of questions in the multiple choice, short response and free response will change from exam to exam and the amount of points that each one is worth will change from exam to exam. So again, focus, you know, go to the exam syllabus to get that information. I want to focus on the types of questions we're gonna see going forward for this exam. Now, this exam is going to cover the materials from the start of the semester up until integration by parts. So that should be lectures one through 10 that'll be covered on this exam. Now, lecture 10 also introduced trigonometric integrals, how we can use trigonometric identities to help us calculate anti-derivatives of trigonometric functions. While that was covered in lecture 10, those topics will not be covered on this exam. This exam will only go through integration by parts. That's not to say that you won't do any calculations with trigonometric functions. It just means you don't need to know lots of trigonometric identities to be successful on this exam. In the multiple choice section, there's gonna be four questions in the multiple choice section for this exam. Each question is worth six points out of 100. And like I said, you get all six points by selecting the correct answer and you get none of them if you don't select the correct answer. So be cautious about that. With question number one, you're gonna be asked to evaluate a definite integral, you know, something like this. The integral from negative two to two of three X plus one squared. You wanna compute the integral. Most likely the tool you're gonna use here is used the fundamental theorem of calculus. The fundamental theorem calculus tells us that if we wanna calculate a definite integral, we can evaluate first an anti-derivative of the function and then you evaluate the function at one or at two and negative two, the bounds in this situation. That's most likely what you would use. So like lecture one, discuss these topics here for us. But be aware there are other things you could use well. I mean, if you wanted to, you may be able to use a U substitution given that this is a square. If you don't wanna foil it out, you could use a U substitution here. That would be appropriate like we did in lecture two. I should also mention that since the top and bottom bounds are the same number, symmetry might be appropriate. If this is an even function, this would be the same thing as two times integral from zero to two of three X squared plus one. I'm not saying this is an even function. I'm saying if it was an even function, you could do that. But if it was an odd function, it's even better. It would just be zero, right? And so the answer could be A, because if you integrate an odd function on a symmetric interval, you always get back zero. So those are techniques that could be useful on question number one, calculate a definite integral using techniques that we saw in lectures one and two. Those techniques from calculus one. So that should definitely be on the easier side. Question two is gonna be a question about work, right? We saw lots of different types of work problems. Most likely this question will probably have something to do with Hooke's law. Hooke's law was this says that the distance that you express, that you extend a spring is proportional to the force required there. So the force to stretch a spring, to hold a spring, I should say extended, is gonna be some constant times X the distance. To calculate work, we probably wanna integrate Hooke's formula right here. And so we saw many examples like that in the lessons in the assignments. So you wanna prepare like something like that. So this will be an easier one. We saw examples where it's like, oh, the force, the force to move an object is like two over X square or something like that. In which case, the thing to know about work, work is the integral of force with respect to distance. And that'll get you through question two fairly well. You might wanna practice some of these Hooke's law questions that you see on the screen right now. But this will be a fairly simple work one. I say it's an easy work problem because in the free response we will see a, dare I say, hard work problem by comparison. Question number three is gonna ask you to compute the average value of a function. So you're given a function in this case, it's F of X equals X squared, you're given an interval. The thing to remember about average value is that the average value of a function is equal to the length of the interval, one minus BA times the integral from A to B of F of X DX. That gives us the average value of the function. We learned how to do that. Let's see, where would that have been? That would have been in like lecture eight, I think, because work was lecture seven and eight. So the second half of lecture eight was about average value. So take a look at some of those if you want some more practice, lecture eight right here. This one will probably come from lecture seven, some of those easier work problems. Question number four is gonna be a question where you're asked to find the area bounded between two curves. In this case, I do happen to use trigonometric functions. Like I said, you do need to know how to do some trigonometric integrals. You just don't need any of the fancy identities that we saw at the end of lecture 10. But this one wants you to find the area between two curves for which we'd integrate from A to B where the bounds depend on the region in play here. You can have some function F of X minus G of X. So one function on top, one function on bottom there, integrate with respect to X. Now, some things we should be aware of. The bounds might be given like in this one, but the bounds might also be implicit. You might have to look at intersections to find where they are. There are also concerns about maybe the functions cross each other. And so who's on top and who has bottom breaks apart at this point. I should mention that on this exam, there's gonna be two questions about finding the area between two curves. The number four in the multiple choice section will be the easier one. There will be a quote, harder one. I mean, it's more moderate than hard, but there'll be a harder one, which we're gonna find in the short response section. You can go to lesson three to learn some more about area between curves. And that then ends the multiple choice section. In the short response section, we also have four questions, I believe. Yep, five through eight. And these times, they're each gonna be worth seven points. So they are a little bit harder than the multiple choice and seven points. But also, unlike the multiple choice, you can get partial credit for partially correct answers. So even though they're worth more points, it's much easier to get at least some of the credit if not all of the credit. Question number five is gonna ask you to calculate an anti-derivative. Most likely for this one, you're gonna wanna use integration by parts, which we learned about, of course, in lectures nine through 10. Remember, integration by parts is very, very useful when you have a factorization of some kind like X times E to the negative X. So remember with integration by parts, if you're ever in the situation where you're integrating a function U with respect to the differential DV, this is equal to UV minus the integral of VDU. And so this is the situation where you're like, okay, I have a function, which I'm gonna find the derivative of. I have a function I'm gonna find the anti-derivative of and you can switch things around. This should be a fairly straightforward integration by parts problem. I should mention that in the free response section, there will be another question which you're gonna need to use integration by parts. That one will be more involved, more challenging. So you can consider this one the easier integration by parts. There will be another one that shows up on this exam. Now, with the show response, the idea is there is a line provided. Please put your answer there. That would be very helpful. There is of course some space provided if you need to do some calculations. You do not have to fill anything in this gap, but it is provided to you so that if you do want to show your work, because there's a few things that to do on this one, like you have to decide who's the U and therefore who's DU, who's the DV there's lots of work you could show on this one. So even if you don't get it fully right, you can definitely get some partial credit on these type of questions. Then we come to question number six for which much like question number five, you'll be asked to evaluate an indefinite integral which means you're looking for an anti-derivative. Regrettably, there's not a lot of space provided but this one should be a fairly simple one comparison. That is you should be able to use very simple techniques to find anti-derivative. So things like we did in lesson one, like for example, you hopefully should be able to find an anti-derivative of cosecant times cotangent. If you don't know what that is, you might want to look it up. That's a fairly standard one. Could you do an anti-derivative of two E to the two theta? So yes, sure, maybe a simple U substitution could be involved there, but these should be fairly simple ones. Like the power rule might also be relevant here. And remember the power rule for anti-derivatives is different versus derivatives, right? If you're integrating the function, the power function x to the n dx, the anti-derivative is gonna equal x to the n plus one over n plus one plus a constant. With, of course, note here that n does not equal negative one and that situation, if you have like one over x dx, the anti-derivative is the natural log of the absolute value of x, like so. Now, with questions five and six, you are asked to evaluate indefinite integrals, which means you're looking for anti-derivatives. In order to get full points on these questions, you do need to make sure you have the plus C. As we calculate indefinite integrals, it is necessary that you have that arbitrary constant. If you do not have that plus C, you're gonna forfeit a point out of that seven points on these two questions. So don't forget the plus C. The next two questions for the short response section, these are gonna be questions that ask you to set up an integral. You will set up, but you will not evaluate the integral. So the answer to this question will be an integral of some kind, okay? So things I would look for is that it's a definite integral. Both of these are definite integrals. So there should be bounds. There should be some numbers, A and B, specific numbers, A and B on the top and bottom of that integral symbol. You're gonna have an integrand, which will be a function, which will depend on the problem and play. You also need a differential. A lot of people skip out on the differential, but it's an important part of the integral. There should be a dx or a dy, whatever is appropriate. If your differential is missing, you've lost some points on this question. If your bounds are missing, it's no longer a definite integral. It's an indefinite integral and that's not correct. So you need to make sure you have bounds that are correct. And then likewise, the function itself should be there. That sort of goes without saying by a set anyways. And then also, you clearly need an integral symbol. You're not gonna write an answer like that. That's just a function. It should be an integral. So you need all the ingredients of the integral here, including the differential and the bounds. That's true for question eight as well. Now on question number seven, I mentioned earlier how it was question number four, you have to do area between two curves. Question number seven is gonna do that as well. So again, we have some area between curves. Now, as opposed to question number four, which we saw in the multiple choice, this one is gonna be a little bit more moderate, a little bit more difficult than the easier one we saw on question number four. Like for example, on question number four, the bounds were given to us, but on this one it wasn't. So as the bounds were implicit, you're gonna have to set these things equal to each other and solve. That's one thing that could be different. This idea of the lines crossing each other in the middle. And so you might have to separate it into two separate integrals. That also might be something necessary for question number seven. It will be, I'm not saying number seven is a hard question, but I'm saying it's gonna be harder than what we saw in question number four. And again, go to lesson three, if you wanna get some more practice on area between two curves. Question number eight, which we also see on the screen right here, this is the last question from the short response section. You also will set up an integral, but you will not evaluate it. This integral will ask you to find the volume of a solid of revolution for which a region will be described to you. So you know which region you're gonna be rotating and access will also be provided to you. These regions, much like we did with the area problems beforehand, the bounds might be explicitly given, like in this case, x equals zero to x equals pi, but it could also be like the previous problem where the bounds are implicit, for which you might have to solve for them by solving some equations, find the point of intersections. Drawing a picture is a good way of showing your work. Draw a picture, I recommend for that. For this one, for this one, and any other one where it should be appropriate. You can get some partial credit by drawing the picture. It helps not just get points, but also helps you understand the problem a lot better. Draw a picture of the region. Also, you have an axis you have to rotate around. It could be the x-axis, it could be the y-axis. It could be any axis. It could be x equals one. It could be y equals seven, the axis of rotation. Be prepared for any of these. But question number eight will be a solid of revolution. And as such, should you use the washer method or should you use the shell method? In all reality, on this test question, you get to pick the one you wanna do. You can use the washer method, you could use the shell method. Many of these problems can be done with either one, both of them measure the volume of a solid of revolution. Although generally speaking, one will be dramatically easier than the other. So I'm not gonna say anything in this video right now, because I'm only talking about the structure and the topics of the exam. I don't actually wanna answer these questions from the practice exam in this video. There are separate videos you can find on Canvas for solutions to any of these problems if you get stuck on them. So look for those if you do have any questions. So I'm not gonna say right now is the washer or shell method better. But what I can tell you is that you are allowed to use either one. And if you use the washer, even if that's the worst method, you can still get full credit if you have the correct integral. You honestly don't have to evaluate the integral. You just have to set it up. But even still, some of them are difficult to set up one versus the other. So part of this question is your judgment. Do you wanna use the washer or the shell method? So the washer method we first introduced in lecture four with the disc method, which is a special case of that. We did some more examples of the washer method in lecture five, and then in lesson six is when we introduced the shell method. So if you want some more practice on these volume problems, go to lessons four through six in our lessons. So that was the last question from the short response section, now we're reaching the free response section. These questions have variable points based upon their difficulty. And so do remember, well, if you wanna know, just look at the points we'll say right there. So I think the smallest one on this exam is worth 10 points, but most of them are in the 12 point range. Much like the short response, you can get partial credit on these questions, but unlike the short response, you must show all your work to get any credit. So show your work to get credit on these ones. But in some regard, there are similar to short response, but they're gonna be a little bit more challenging. Question number nine, you'll be asked to evaluate an indefinite integral, which means look for a anti-derivative, but don't forget the plus C. You do need that with indefinite integrals here. With this one, you probably wanna use a U substitution on question number nine. I'm not gonna tell you what the U is necessarily, but you should declare. One of the ways of showing your work is you're like, oh, U equals this function, F of X, D U then equals F prime of X, D X. You then rewrite this thing as something like, oh, it's now D U over U to the one-third, maybe. So that's then U to the negative one-third, maybe. Cause there could be other things, like maybe there's some coefficients you have to change. Maybe I need like a two in here. There's things you have to do here, but so number nine, do anticipate a U substitution that you'd have to do. You probably wanna find the anti-derivative with respect to U. Then you need to switch it back to X before you're done. U substitution, we reviewed in lesson number two. So go to that lesson and the associated assignments connected to lesson number two to get some more practice on U substitution. Question number 12 is gonna be another question about integration by parts like we saw before. It's an indefinite integral. You're only looking for the anti-derivative. You do need a plus C at the very end for full credit on this one. But unlike the U substitution we saw in question number nine, you wanna use integration by parts. We did spend two lectures, two lessons on integration by parts. These were lessons nine and 10. Some of the harder ones we saw at the end of lesson nine at the beginning of lesson 10. Those are the ones you should expect for number 10, this question here. So while the previous question was question number five, which should be a fairly straightforward integration by parts, number 10 will be a little bit more involved. Like some of the variations like integration by cycles or integration by hope might be appropriate tools to use on these ones. Question number 10 might use integration by parts but combines U substitution as well. So again, number 10 you should think of as the most advanced anti-derivative that you have to calculate on this exam because it combines all of the techniques we've learned about for the course up until this moment. On the last page of the exam, we have two more questions. Questions 11 and question 12. So there's four questions in the free response section. These are both worth 12 points, I believe. This one definitely is worth 12 points. What you're gonna see on the last page of the exam is two applications using integrals. So like in the show response section when we had questions seven and eight, you were gonna set up an integral but you were not gonna evaluate it. Even though this is a free response question, you're just gonna set up the integral. But unlike the ones in the show response, there's a lot going into these problems. So there's gonna be things to draw, there's gonna be equations to solve, there's gonna be substitutions to make. There's a lot of stuff going on that you can show your work. Question 11 is gonna be one of these problems where we have a two-dimensional base that we then are gonna stack cross sections on it. So we might stack rectangles on top of a circle or we might put triangles inside of a parabola. These were type of things that we did in lesson five, I believe was the number. Because lesson five started with some more washer method type problems, but then the second half of that assignment, sorry, the second half of that lesson in its associated assignment dealt with these sort of cross-sectional volume problems. So not volumes by revolution. The idea is the volume is the integral of the area function where A of X is the cross-sectional area of a slice. And then DX is the thickness of such a slice. You'll have some bounds. On this question, number 11, you were setting up an integral but not evaluating it. In order for full credit, you need to have the correct function, that's where most of our attention goes to, but you should have the correct bounds. So make sure you show your work to know what those bounds are and don't forget the differential. For full credit, you do need that differential. And again, there's some examples of this from lesson five. The last question on the exam is also worth 12 points. This is gonna be a work problem, but this one's gonna be more involved in what we saw in the multiple choice section. Remember, question number two was supposed to be an easy work problem. This one's definitely gonna be a more difficult work problem. So this will be things like, we're gonna pump some water out of a tank, which is the type of that problem you see on the screen right now. We have things like lifting a rope. We had other problems like a leaky bucket. That is you're lifting something whose weight is changing over time. And so how do you deal with that? We did some examples like this. We saw examples like this in lesson seven and eight. We did the rope problems and the leaky bucket problems in lesson seven and in lesson eight, we did the water pumping problem. The second half of lesson eight was about average value, which you won't see on this one here. Much like question number 11, you're gonna set up and solve these ones. But sorry, you're gonna set up but not evaluate the integral. Just set up the integral. For full credit, you need all of the ingredients of the integral, including the differential. And ways to show your work is much like you would on question number 11. Draw pictures, solve equations, because you might have to solve some equations. Determine the bounds. Look at the slices you do and work through it. There's a lot going on into each of these problems. So you show all of that onto the screen. And that gives us the last question here on the exam. Now, if you've been adding up the scores, you notice this adds up to be 98 points. The last, there is officially a question 13 on the exam, but this is just a reminder to turning your study resource. Typically this is gonna be a three by five note card for which look to the exam syllabus to see which type of study resource are you allowed to use on the exam. That will be submitted with the exam. And there's two points out of 100 for that study resource. Like again, go to the exam syllabus to see exactly what study resource is allowed for this exam. But for all of the exams we look at this course, there's always two points at the very end for that study resource, which I will always just refer to as your note card, okay? If you have questions on calculators or can you use scratch paper? What's other study resources can I have? Please consult the exam syllabus or just ask your questions. Cause again, these can change from time to time. So that's not embedded inside of this review. I want to mention that in order to get a good grade on these exams, you want to dedicate a substantial amount of time. I've put a lot of thought into this. And honestly, I think that to get a passing grade on the exams in this class, you are going to probably want to dedicate at least four hours of studying towards this exam. And there's a lot of study resources available. Honestly, watching this video, which is about 30 minutes in length, is part of your studying for this exam, looking at the practice exam, working through the practice exam, reading and working through the exam syllabus. These are also time that goes towards this exam. It takes a lot of time to get good at these cyber problems and they're worth a lot of points. So you want to be good at them. Understand that. And I say this not to scare you, but to set an expectation. If you're expecting to get a good grade on this exam and put little to no effort into studying, you probably will be wrong. Some of you might be fantastic and you're already amazing at this stuff and you won't need any time at all. That might be the case, but that's not the typical student. And so if I'm talking to you, you probably are a typical student and you probably need some time. And it's gonna be hours to prepare for this exam. Plan ahead and give yourself the time so that you can take this exam. And of course you also need time to take the exam itself. And so that brings us to the end of this review video. Like I said, there are other resources on Canvas that you should take a look at if you haven't already. There are the exam syllabus. There is, of course, the practice exam. There's the solution videos to practice exam and some other resources that you should find on Canvas. Dive into those. Let me know if you have any questions. I am here to help you do well on this exam. And if you put in the effort and get the support that you need, you will do well on this exam. And so with that bit of hope, I will talk to you next time. Bye.