Volumes with cross sections: squares and rectangles
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Volume with cross sections: squares and rectangles (no graph)
- [Instructor] The base of a solid is the region enclosed by the graphs of y is equal to negative x squared plus six x minus one and y is equal to four. Cross sections of the solid perpendicular to the x-axis are rectangles whose height is x. Express the volume of the solid with a definite integral. So pause this video, and see if you can have a go at that. All right, now what's interesting about this is they've just given us the equations for the graphs, but we haven't visualized them yet. And we need to visualize them, or at least I like to visualize them so I can think about this region that they're talking about. So maybe a first thing to do is think about, well, where do these two lines intersect? So when do we have the same y-value? Or another way to think about it is when does this thing equal four? So if we set them equal to each other, we have negative x squared plus six x minus one is equal to four. This will give us the x-values where these two lines intersect. And so we will get, if we want to solve for x, we can subtract four from both sides. Negative x squared plus six x minus five is equal to zero. We can multiply both sides by negative one. We will get x squared minus six x plus five is equal to zero. And then this is pretty straightforward to factor. One times five is five, or actually I say negative one times negative five is five. And negative one plus negative five is negative six. So it's going to be x minus one times x minus five is equal to zero. And so these intersect when x is equal to one or x is equal to five. Since we have a negative out front of the second-degree term right over here, we know it's going to be a downward-opening parabola. And we know that we intersect y equals four when x is equal to one and x equals five. And so the vertex must be right in between them, so the vertex is going to be at x equals three. So let's actually visualize this a little bit. So it's going to look something like this. I'll draw it with some perspective 'cause we're gonna have to think about a three-dimensional shape. So that's our y-axis. This is our x-axis. And let me draw some y-values. So one, two, three, four, five, six, seven, eight. This is probably sufficient. Now we have y is equal to four, which is going to look something like this. So that is y is equal to four. And then we have y is equal to negative x squared plus six x minus one, which we know intersects y equals four at x equals one or x equals five. So let's see, one, two, three, four, five. So x equals one, so we have that point right over there. One comma four, and then we have five comma four. And then we know the vertex is when x is equal to three, so it might look something like this. We could substitute three back in here. So let's see, y will equal to negative nine, three squared, plus 18 minus one. And so what is that going to be? That's going to be y is going to be equal to eight. So we have the point three comma eight. So this is five, six, seven, eight, yep, right about there. And so we are dealing with a situation, we're dealing with a situation that looks something like this. This is the region in question. So that's going to be the base of our solid. And they say cross sections of the solid perpendicular to the x-axis, so let me draw one of those cross sections. So this is a cross section perpendicular to the x-axis, are rectangles, whose height is x, so this is going to have height x right over here, height x. Now what is this, the width, I guess we could say, of this rectangle? Well, it's going to be the difference between these two functions. It's going to be these, this upper function minus this lower function. So it's going to be, that right over there is going to be negative x squared plus six x minus one and then minus four, minus the lower function. So that could be simplified as negative x squared plus six x minus five. And so if we wanna figure out the volume of this little section right over here, we'd multiply x times this, and then we would multiply that times an infinitesimal small, infinitesimally small depth, dx. And then we can just integrate from x equals one to x equals five. So let's do that. The volume of just this little slice over here is going to be the base, which is negative x squared plus six x minus five, times the height, times x, times the depth, times dx. And then what we wanna do is we wanna sum up all of these. So you could imagine right over here, you would have, or like right over here, you would have a cross section that looks like this. X is now much larger. The height is x, so now it looks something like this. So I'm just drawing two cross sections, just so you get the idea. So these are the, this is any one cross section for a given x, but now we want to integrate. Our x is going from x equals one to x equals five, x equals one to x equals five. And there you have it, we have expressed the volume of that solid as a definite integral. And it's worth noting that this is, that this definite integral, if you distribute this x, if you multiply it by all of these terms, it's very solvable. You don't need a, or it's very solvable without a calculator. You're just going to get a polynomial over here that you have to take the antiderivative of in order to evaluate the definite integral.
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