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Current time:0:00Total duration:12:53

Video transcript

so let's say I have a function of x and y f of x and y is equal to x plus y squared if I try to draw that let's see if I can have a good attempt at it that is my y-axis I'm going to little perspective here this is my x-axis I can make it do the negative x and y-axis you could do in that direction but this is my x-axis here and if I were to graph this when y is 0 it's going to be just a let me draw it in yellow it's going to be just a straight line that looks something like that and then for any given X you're going to have a parabola in Y right Y it's going to look something like Y it's going to look that I'm just going to do it the positive quadrant going to look something like that like that like that it'll actually when you go into the negative Y it's kind of going to see the other half of the parabola but I'm not going to worry about it too much so you're going to have this surface that looks something like that maybe I'll do another attempt at drawing it but this is our ceiling we're going to deal with again and then I'm going to have a path I'm going to have a path in the XY plane I'm going to start at the point I'm going to start right there at the point 2 comma 0 X is equal to 2 y 0 and I'm going to travel just like we did in the last video I'm going to travel along a circle but this time the circle is going to have radius to move counterclockwise in that circle this is on the XY plane just to be able to visualize it properly so this right here is the point 0 2 then I'm going to come back along the y axis this is my path I'm going to come back along the y axis just like that and then I'm going to cut take a so I took a left here and then I'm take another left here and then come back along the x axis I'm going to come back along the x axis right like that so this what I driven these two shades of green that is my contour and what I want to do is I want to evaluate the surface area of essentially this little building that has the roof of f of X Y is equal to X plus y squared and whose and I want to find the surface rivets walls so they'll have this wall right here whose base is the x-axis then you're going to have this wall which is along the curve it's going to look something like you know I don't it's going to look kind a funky wall on that curved side right there I'll try my best effort to try to actually is going to be curving way up like that so it's going to be curved up like that and then along the y axis along the Y X when X is equal zero is going to be a probe it's going to have like 1/2 of a power parabolic wall right there I'll do that back wall along the y-axis I'll do that in orange actually I already using oil I'll use magenta that is the back wall along the y-axis then you have this front wall along the x-axis and then you have this weird curvy curvy curtain or wall that may be in blue that goes along this curve right here this part of a circle of radius 2 so hopefully you get that visualization it's a little harder I'm not using any graphic program this time but I want to figure out the surface area the combined surface area of these three walls and we in very simple notation we could say well the surface area surface area of those walls of this wall plus that wall plus that wall is going to be equal to the line integral the line integral along this curve or along this contour however you want to call it of f of X Y so that's X plus y squared DS where D s is just a little length along our contour and since this is a closed loop we'll call this a closed line integral and you'll sometimes see this notation right here that notation often you'll see that in physics books and well we'll be dealing with a lot more and you'll put a circle on that integral sign and all that means is that the contour we're dealing with is a closed contour we get back to where we started from but how do we solve this thing well a good place to start is to just define the contour itself and just to simplify it we're going to divide it into three pieces and essentially just do three separate line integrals because these you know this isn't a very continuous contour so the first part let's do this first part of the curve where we're going along every a circle of radius two and that's pretty easy to construct if we have X if we have X is equal to let me do that in each part of the contour in a different color so if I do an orange this part of Conte or if we say that X is equal to two cosine of T and Y is equal to two sine of T and if we say that T and this is really just good building off just what we saw in the last video if we say that T and that this is from T is greater than or equal to zero and is less than and less than or equal to PI over two T is essentially going to be the angle that we're going along this circle right here and this will actually describe this path and if you know how I constructed this a little confusing you might want to review the video on parametric equations so this is the first part of our pad so if we just wanted to find the surface area of that wall right there we know we're going to have to find DX DT and dy DT so let's get that out of the way right now so if we say DX DT is going to be equal to minus two sine of T dy DT is going to be equal to two cosine of T just the derivatives of these we've seen that many times before so if we want this orange walls surface area we can take the integral and if any of this is confusing there are two videos before this where we kind of derive this formula but we could take the integral from T is equal to 0 to PI over 2 of our function of X plus y squared X plus y squared and then times the D s times so X plus y squared will give the height of each little block and then we want to get the width of each little block which is D s but we know that we can rewrite D s as the square root as the square root I'll giving me some give myself some room right here of DX of the derivative of X with respect to T squared so that is minus 2 sine of T squared plus the derivative of Y with respect to T squared so plus 2 cosine of T squared DT this will give us the orange section and then we can worry about the other two walls and so how can we simplify this well this is going to be equal to this is equal to the in all right over here the integral from 0 to PI over 2 of X plus y squared and actually let me write everything in terms of T in terms of T so X is equal to 2 cosine of T so let me write that down so it's 2 cosine of T plus y which is 2 sine of T plus 2 sine of T and we're going to square everything and then all of that all of that times this crazy radical right now it looks like a hard antiderivative or integral to solve but I think we'll find out it's not too bad this is going to be equal to 4 sine squared of T plus 4 cosine squared of T we can factor a 4 out now we don't want to forget the DT this over here let me just simplify this expression so I don't have to keep rewriting it that is the same thing as the square root of 4 times sine squared of T plus cosine squared of T we know what that is that's just 1 so this whole thing just simplifies to the square root of 4 which is just 2 so this whole thing simplifies to 2 which is nice first solving our antiderivative simplifies things a lot so this whole thing simplifies down to I'll do it over here now I don't want to waste too much space I have two more walls to figure out the integral from T is equal to 0 to PI over 2 and I want to make it very clear I just chose the simplest parametrizations I could for X&Y but I could have pick two other parametrizations and I would have had to change T accordingly so as long as you're consistent with how you do it it should all work out there isn't just one parametrizations for this curve it's kind of depending on how fast you want to go along the curve but watch the parametric video parametric functions videos if you want a little bit more depth on that but anyway this thing simplifies we have a two here two times cosine of T that's 4 cosine of T and then here we have 2 sine squared sine of T squared so that's 4 sine squared of T so it's for sine squared of T and we have to multiply it times this two again so that gives us an eight eight times sine squared of T DT and then you know sine squared of T that looks like a tough thing to find the antiderivative for but we can remember that sine squared of really anything like we could say sine squared of U is equal to 1/2 times 1 minus cosine of 2-u so we can reuse this identity if you know we I can try to t here sine square root of T is equal to 1/2 times 1 minus cosine of 2t let me rewrite it that way because that'll make the integral a lot easier to solve so we get the integral from 0 to PI over 2 and actually I could breakup well I won't break it up a 4 cosine of T for cosine of T plus 8 times this thing right 8 times this thing this is the same thing as sine squared of T so 8 times this 8 times 1/2 is 4 4 times 1 minus cosine of 2t just use a little trig identity there and all of that DT now this should be reasonably straightforward to get the antiderivative of let's just take it the antiderivative of this is antiderivative of cosine of T that's just sine of T right the derivative of sine is cosine so this is going to be 4 sine of T the scalars don't affect anything and then well let me just distribute this 4 so this is 4 times 1 which is 4 minus 4 cosine of 2t so the antiderivative of 4 is 4 T plus 4 T and then the antiderivative of minus 4 cosine of 2t let's see it's going to be sine of 2t let's see the sine of 2t sine of 2t the derivative of sine of 2t is 2 cosine is 2 cosine of 2t we're gonna have to have a minus sign there and put a 2 there and now it should work out what's the derivative of minus 2 sine of T take the derivative inside 2 times minus 2 - four and the derivative of sine of 2t with respect to 2t is cosine of 2t so there we go we've figured out our antiderivative now we evaluate it from 0 to PI over 2 0 to PI over 2 and what do we get we get 4 sine of Pi let me write this down 4 I don't wanna skip too many sine of PI over 2 plus 4 times pi over 2 that's just 2 pi minus 2 sine of 2 times pi over 2 sine of PI and then all of that minus all of that minus all of this evaluate at 0 that's actually pretty straightforward because sine of 0 is 0 T moving 4 times 0 is 0 and sine of 2 times 0 that's also 0 so all everything are the zeros work out nicely and then what do we have here sine of PI over 2 is in my head I think sine of 90 degrees same thing that is 1 that is 1 and then sine of pi is 0 that's 180 degrees so this whole thing cancels out so we're left with 4 plus 2 pi so just like that we were able to figure out we were able to figure out the area of this first curvy wall here and frankly that's the hardest part now let's figure out the area of this curve and actually going to find out that these other curves since they go along the axes are much much much easier but we're going to have to find different parametrizations for this so if we take this curve right here if we take this curve right here let's do a parametrizations for that actually you know what let me continue this in the next video because I realize my videos have been running a little long but I'll do the next two walls and then we'll sum them all up