Calculus problem

mysticdemon

OK, so in my calculus class i have a final project to do. I have to figure out the approximate volume of a dome using calculus. I have to use some kind of calculus and tell whether i think the estimate is bigger or smaller than actual. Then i have to write a 2-5 page essay telling the things I tried to use and my eventual method. Anyway the dome i have to find the volume of is a tent that measures 96inx96in is the base. It is a square and its height is 55 in.

Ive tried using a few different methods like measuring under a curve with the antiderivative of pi r squared, the washer method/shell method, and just trying to make a cylinder, pyramid, or rectangle using the base and seeing how close i can get.

Ok, so I was wondering if anybody could give me any ideas or answers or just any input on what you think about how to solve this problem. Thank you in advance.

Serpent

i would probably just integrate over the square slices. your answer will be wrong by the amount of dome that bulges outside of this, based on the curvature between corners.

MrOletta

Could you maybe be a little more clear on the geometry of the figure? Perhaps an artistic attempt in paint.

Big Dookie

Just to be clear, when you say "dome", what exactly does this mean? I see that you want the volume, which means it must be a 3-Dimensional character (which seems strange to me if this is a Calculus 1 class, since you generally don't get into 3-Space and Multivariable Calculus until the second or third semester). But what does the dome look like exactly? Is it a short Cylinder with a rounded top, like the Astrodome, or is it just a half-sphere?

Edit - Wait, nevermind, I just saw the part about the tent and square and everything. However, I'm still a little confused, because that doesn't sound much like a dome. Like MrOletta said, some kind of general picture would help a lot.

mysticdemon

Well, here is my attempt at paint for the tent. It looks kind of like a pyramid im not a very good artist but its basically that just a bit more rounded at the top.

Big Dookie

Does it come to a definite point at the top, or is it completely rounded off at its tip?

Sorry, I know it's kind of specific, but it could make a difference.

Edit - I'm at work, so I can't really work on this much right now, but here's something to think about: when you are finding a 2-dimensional integral on the x-y plane, you're basically cutting it up into an infinite number of slices and adding up the height of all of those slices, right? So in a way, you could think of it as an infinite number of lines that have no"width", and you're adding up the height of all of those lines.

Now, apply that same concept to an object in 3-dimensional space. Conceptually, think about it the same way as the other one. The first thing you would do is cut up the function into an infinite number of "slices", so that the entire thing is now composed of an infinite number of functions that all like in the 2-dimensional plane. You would take the integral of "all" of those functins (impossible in reality, since there are infinitely many of them, but it is very possible mathmatically), and then you would add all of them together.

Mathmatically, this is done by using a double integral. However, I'm assuming you haven't gotten to this yet. In any case, think about this, and think about how it may relate to something like the Riemann Integral. I'm not saying this will lead you to the solution, as I don't know what that is yet, but it may at least get you thinking in the right direction.

AgentBryant

This seems like it would be a problem where you have a surface (the tent) in which a certain cross-sectional area will be like a square or something (I don't know enough about the problem to know what exactly) that would be summed up with a function describing the size of each piece.

It's hard to tell without more information, though.

RocketScience

I imagine the tent looks like this.

The main part of the exercise is to explain how you went about choosing your particular method of approximation. If you were going to go ahead and find the exact volume you wouldn't need 2-5 pages to justify your answer.

Vrtra Theory

I'm guessing his specific tent looks something like this.

Either way, I'll just add on to what Big Dookie mentioned: picture the tent as a series of horizontal slices (for argument's sake, pretend there are 55 slices, each 1 inch high.) Each slice of the tent has a volume of Width * Length * Height - the very top slice might be 0 * 0 * 1, and the very bottom slice is 96 * 96 * 1. The rest of the slices are somewhere in between. If you add them all together, you get a good approximation of the total volume. So, all you need to do is figure out a function that best describes the width and length of a given slice.

For example, if you used a very simple equation for width and height, f(x) = x/55 * 96, with x running from 0 at the top to 55 at the bottom, your equation would describe a pyramid with no curvature at all. The volume of each slice would be f(x) * f(x) * 1.

It looks to me like the tent is close enough to a perfect dome that you could describe the curve of its width and length with sin(x). For example, you might say that the width and length of each slice are both f(x) = sin(Pi/2 * x/55) * 96 - where x runs from 0 at the top to 55 at the bottom. Again, f(x) * f(x) * 1 would be your total volume for each slice.

If the tent isn't really particularly dome-like after all, you might mess around with other curves that look similar (square root might be closer than sine, actually.) The only "rule" to the equation you come up with is that plugging in 0 for x should give 0, and plugging in 55 for x should give 96 (based on the dimensions you gave.) A value in-between should give values in between.

Out of curiousity, what level of calc is this for? Your question doesn't make it clear if you're used to solving these problems with sums or integrals or some other method.

mysticdemon

yes, the tent is rounded at the top. It looks pretty darn close to the tent Vrtra Theory posted. I thought about the slices thing i just haven't had a whole lot of time to figure it totally out. I am in a high school calc class if that helps so we probly have just gone over some very basic stuff I'm assuming. I was thinking something like a three-dimensional Lram, Rram, or Mram thing but i kind of messed up on that. Thanks for your input i will look check 'em out to see if i can figure this thing out.