Parentheses on this whole thing, I want to then square it. Six minus four times the natural log of x, or actually the natural We're going from zero till two of, and then we have, let me open parentheses 'cause I'm gonna have Nine for definite integral, and then we just have to input everything. And so, we can hit math and then hit choice number Integral to evaluate by hand, but we can actually useĪ calculator for that. And if you were to evaluate this integral, you would indeed get the volume of this, this kind of pedestal horn-looking thing. Three-dimensional shape, as opposed to just the height This three-dimensional shape, the surface area of this And instead of just multiplying dx times the differenceīetween these functions, we're going to square theĭifference of these functions 'cause we're visualizing This thing right over here would be this thing right over here, where it's dx. Intersect our base, you would say all right, So we could just integrate from x equals zero to x equals two, from x equals zero to x equals two. To add up all of these from x equals zero to x equals two? Well, then you would have the Of just this little section right over here, and I think you might And then you multiply it times the depth. It is going to be six minus, our bottom function is four times the natural log of three minus x, and so that would just What's the base length? Well, it's the differenceīetween these two functions. Pink right over here? Well, that area is going toīe the base length squared. So what would be the area that I am shading in in Of one of these things? Well, it would be theĭepth times the area, times the surface area of this cross section right over here. That have a very small depth, that we could call dx. You could view it as a, break it up into these things You could even, I couldĭraw it multiple places. Little tile, this one into it, that also has some depth to it. Have to break up the shape into a bunch of these, youĬould view them as these little square tiles that have some depth to them. That you already have the powers of integration to solve this. These three dimensions? But you'll quickly appreciate Well, hey, I've been dealing with the two dimensions for so long. And some of you might be excited, and some of you mightīe a little intimidated. Little bit so that you can appreciate it a little bit more, but hopefully you get the idea. And so the whole shape would look, would look something like this, would look something like that, try to shade that in a We can draw the whole thing, roughly at the right proportion. It's going to be quite big, might have to scroll down so This length, which is six at this point, this is also going to be the height. Whatever the differenceīetween these two functions is, that's also how high we are going to go. We also go that much high, and so the cross section isĪ square right over there. Section right over here, is going to be a square. Where any cross section, if I were to take a cross And so this region is this region, but it's going to be the base of a three-dimensional shape And then the graph of y is equal to four times the natural log of three minus x would look something like this, look something like this. Just draw it like this, and so this would be This is the line y is equal to six right over there. So let me draw this over again, but with a little bit of perspective. Gonna think about a shape, and I'm gonna draw it in three dimensions. We're gonna find the volume of shapes where the base is defined in some way by the area between two curves. But what we're going to do in this video is do something even more interesting. For example, we couldįind this yellow area using a definite integral. And, in fact, if you're not, I encourage you to review Likely already familiar with finding the area between curves.
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