Determining integral for volume with rectangular cross sections
About this video
The base region R lies between f(x) = 3cos(pi x/8), g(x) = (1/4)x - 1 and the y-axis, and the curves meet at (4, 0). Cross sections perpendicular to the x-axis are rectangles with base f(x) - g(x) and height 3x. Volume = integral from 0 to 4 of [3cos(pi x/8) - (x/4 - 1)](3x) dx.
You'll be able to
- find the bounds of a base region from where the curves meet and the y-axis.
- write the volume of a rectangular slice as base times height times dx.
- set up the definite integral for a solid with rectangular cross sections.
Key words
- Rectangular cross section
- A slice of the solid shaped like a rectangle, with area base times height.
- Perpendicular to the x-axis
- At a right angle to the x-axis, so each slice stands straight up at one x-value.
- Slab
- A thin slice with depth dx; its volume is cross-section area times dx.
Transcript
- [Narrator] So we are told that f of x is equal to three cosine of Pi x over 8. This is f of x right over here, and g of x is equal to 1/4 x minus 1. g of x is in this green-ish color. Let R be the region enclosed by the graphs of f and g in the y-axis. So, R is this region right over here.
Let me keep it clean though. Region R is the base of a solid. For each x value, the cross section of the solid taken perpendicular to the x-axis is a rectangle whose base lies in R and whose height is 3x. What definite integral gives the volume of the solid? So we just need to set up the integral.
We don't actually need to evaluate it, but pause the video, try to do it on your own, and then I will do it with you. Okay, I always just like to visualize what we're trying to do, and it's a little harder 'cause this is in 3D, but let me set it up like this so I can, whoops.
So let's say that this right over here is the y-axis. That's the y-axis. This over here is the x-axis, x-axis. If R looks something, not R, g of x will look something like this. So let me see if I can draw it. Drawing everything in an angle here. So that is g of x, it's a line, and then f of x, at least in the region that we care about, let me get a red color, is looking something like that.
And this looks like it's happening at x equals 4. And let's actually confirm. It looks like they both intersect at X equals 4. Both of 'em are equal to zero. But let's confirm that. If I take X equals 4, 4 pi over eight is the same thing as pi over 2. Cosine of pi over 2 is indeed equal to zero.
So f of 4 is zero, and 1/4 times 4 is 1, minus 1 is indeed zero. So yes, both f, this point where they intersect is indeed the point 4,0. Both of them when evaluated at 4 are equal to zero. Let's see if we can visualize this actual solid. So for each x value, so let me pick an x value here, and I'll do it in a light color.
So let's say I had an X... That's too light. Let's say I had maybe an orange. So let's say I had an x value here. The cross-section of the solid taken perpendicular to the x-axis. So the solid, so if I take a cross section perpendicular to the x-axis, so that's perpendicular right over there, it is a rectangle whose base lies in R.
So that's the base that I just drew. And whose height is 3x. So for this one, if this distance right over rear is x, three times X would be something like that. So that's the cross-section there. The cross-section there is something like that. And then the cross-section out here, when X is equal to 4, actually let me go a little bit before that.
It's going to have a very small base, but its height is going to be quite high. It's gonna be approaching 12 there. So it's gonna look something like that. And at x equals zero, three times zero is still zero. So it's gonna look something like this. So it's gonna look something, something like that.
And then of course it's gonna be defined by that curve in the back. That is what the solid is going to look like. Hopefully that helps visualizing it a little bit. But now let's just actually set up the integral. So, one way to think about it is, we could take each of these slabs, and let me give it a little bit of depth.
Let me give each of these slabs a little bit of the depth, and I'll color that in, each of these slabs. And so the volume of this slab is going to be the base here, the length of this base, this is meant to be parallel to the y-axis. So what the length of that base going to be?
That is going to be, I will write it in a neutral color. So that base is going to be f of x minus g of x. That's what that length is. f of x minus g of x. And then what's going to be the height at any of these points? Well that height is going to be 3x, and then we're gonna multiply that times the depth, which is dx.
So that's for any one of them. But if I wanted the volume of the entire figure, I would wanna take the integral. And so I would take the integral from, I'm going from x equals zero to x equals 4, 0 to 4, of the each of these f of x minus g of x times our height, which is times 3x times dx.
Or if I just wanted to actually write it out, it would be the integral from 0 to 4. And actually let me use brackets here, make it a little bit neater. f of x, maybe I'll write that in red. It would be three cosine of pi x over 8 minus, and then g of x I'll do in the green color, g of x is minus 1/4 X minus 1, close that parentheses and then close that bracket, and then times 3x, times 3x XD.
And we are done.
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