# The usefulness of Calculus?

#### saucetray

I am here to ask a few questions I have asked myself and others that have no certain answers to. Now as it is, my love for math is splendid and deep and I even do math problems for fun but I can't help wonder what some parts of calculus are actually useful or I have a question about the intuition.

For example, the volume of a solid of revolution with integration. Sure that's fun and all but how will that ever help in the real world? Sure, if you have math problems or a math test it can help and help derive the geometry formulas for shapes (which I also question their ability to help in the real world as no shape is ever a perfect trapezoid or rectangle--well in rare cases possibility.) I remember a problem in 9th grade geometry about finding the volume of a soda can... Well just because the height is 7 doesn't mean it's that volume.. The top concaves inward... Guessing you could make the top into half a sphere or something and find that volume and subtract but is that ever the true volume fully? In high school I always would look to see if I could find a shape that looked so true and could actually use a formula; for example, when I made a triangle with my phone and 2 books. Sure looked pretty good, did volume of a triangular prism but there was a curve on the corner and side of my phone... One that easily rendered my volume off by hardly a little more than noticeable margin but still bothersome.) anyway... I have a hard time realizing how you can put real world shapes on a xyz coordinate plane and find its volume. Even just the XY coordinate plane... How are you supposed to find the functions to fit each curve and concave, etc...) I find it hardly useful unlike maximum and minimum problems which are quite actually very useful in the math world. I mean I guess you can figure out how to make shapes with certain volumes but figuring out the volume of let's say a mold would be a bit bizarre... Finding 2 functions to exactly mimic it's curve would be impressive and then mathematically sitting at a desk for 1-6 minutes (depending on your ability with math) sounds... Unreasonable and time consuming.. Whether you can say the same about maximum and minimum is also up for debate, sure sitting down with a pencil and paper to find the minimum amount of material you have to use is time consuming. Sure, this may sound ignorant and if you got this far, I am surprised. I'm curious how this applies to the real world. Don't get me wrong, I'm a math major in computer science, not a pure mathematics major. I have been through a lot of calculus and every level of it with A+ but I can't seem to connect it to the real world as well; however I do see the usefulness sometimes. Any thoughts? Am I wrong about something? Please I'm open to anything. This isn't to sound like a ignorant fool. I am curious. A lot of calculus is useful to me, I loved to using it in physics and that was a very useful part of calculus: physics. But some things like solids of integration seem trivial but it's still cool to know.

#### mathman

Forum Staff
Calculus is useful to those who use it.

#### v8archie

Math Team
no shape is ever a perfect trapezoid or rectangle
They don't need to be. It is only necessary to approximate the real world shape sufficiently closely - and all polygons can be divided into triangles.

Mathematics doesn't exactly mirror the real world. We use it to create models that predict the real world to a certain accuracy. One very important part of mathematics is about quantifying the accuracy of estimates and models.

How are you supposed to find the functions to fit each curve and concave, etc...)
There are plenty of techniques for finding functions that approximate given data. Again, the key is knowing how accurate a given approximation is - and again there are techniques for doing this.

sitting at a desk for 1-6 minutes (depending on your ability with math) sounds... Unreasonable and time consuming
In the real world, we usually use technology to work out solutions to real world problems. But a sound understanding of the theoretical basis of the work is necessary in detecting errors and managing the accuracy of the approximations.

Mathematics is a research subject. It's not a science so much as a deductive art. The results of mathematics often have applications in the real world, but the methods are not necessarily applied directly to problems unless done via a computer.

The real world doesn't care if something is 3.14159265359 inches $\pi$ inches. We rarely can or need to measure that accurately.

In essence, you are asking the wrong questions. If I were finding the volume of a container, I would probably measure the volume of liquid required to fill it or look at the manufacturing specification.

#### saucetray

That sounds cool, what kind of ways can you find functions for curves of something in real life?

#### v8archie

Math Team
Well at it's simplest, you can find (or measure) $n$ points on the curve and then use them to create a polynomial of order $(n-1)$ (or greater).

#### studiot

Firstly let me observe that computing is no more (or less) use or real in the 'real' (physical) world than calculus.

Both need applications to achieve that.

I see you are from New York state, where there are plenty of complicated highways so a look at highway engineering would supply answers to both why calculus and why computers and the answer to another question you probably haven't considered viz are there mathematical questions that have no fixed form direct formula solutions in the real world? and indeed there are.

Highways curve this way and that.

The curves are often circular, usually connecting straight sections between.
However what you probably don't know is that highways engineers often introduce what are known as 'transition curves' between sections of highway, to ease the acceleration transition from one curvature to another.
This involves calculus to get these curves right.
There is also curvature in the vertical direction because the land is not flat.
Again calculus is involved generating curves that promote visibility and allow economic earthwoks.
Talking of earthworks, computers (these days) are ideal for performing what are known as cut and fill balance calculations both to reduce construction costs and to calculate the actual volumes of earth to be shifted and the payment therefore.

Does this help?

#### Benit13

Math Team
For example, the volume of a solid of revolution with integration. Sure that's fun and all but how will that ever help in the real world?
For example, to double check whether the output of a computer program designed to calculate the volume is correct. It's also a fairly easy example of calculus in general, which has many uses, so it has pedagogical value.

Sure, if you have math problems or a math test it can help and help derive the geometry formulas for shapes (which I also question their ability to help in the real world as no shape is ever a perfect trapezoid or rectangle--well in rare cases possibility.) I remember a problem in 9th grade geometry about finding the volume of a soda can... Well just because the height is 7 doesn't mean it's that volume.. The top concaves inward... Guessing you could make the top into half a sphere or something and find that volume and subtract but is that ever the true volume fully? In high school I always would look to see if I could find a shape that looked so true and could actually use a formula; for example, when I made a triangle with my phone and 2 books. Sure looked pretty good, did volume of a triangular prism but there was a curve on the corner and side of my phone... One that easily rendered my volume off by hardly a little more than noticeable margin but still bothersome.)
Perhaps the examples you have dealt with can be simplified or approximated by using simpler sets of geometry or equations, but many instances cannot be simplified in such a way. For example, the thermal systems I work with on a daily basis cannot be approximated as such; calculus is fundamental to the solutions we generate and being unable to understand calculus would mean I would never have got past the interview stage when applying for the job!

anyway... I have a hard time realizing how you can put real world shapes on a xyz coordinate plane and find its volume. Even just the XY coordinate plane... How are you supposed to find the functions to fit each curve and concave, etc...)
Many of the examples you have been given in school/college are probably created using basic functions so that you are able to integrate/differentiate. In the real world, we sometimes get 'nice' equations, but more often than not you will have to use some sort of fitting tool and/or choose functions you think will approximate the systems you are dealing with the best. Unfortunately, the functions that can be derived are sometimes very horrible and you may need to use numerical models. However, this should be considered as an additional tool, not a replacement one.

I find it hardly useful unlike maximum and minimum problems which are quite actually very useful in the math world. I mean I guess you can figure out how to make shapes with certain volumes but figuring out the volume of let's say a mold would be a bit bizarre... Finding 2 functions to exactly mimic it's curve would be impressive and then mathematically sitting at a desk for 1-6 minutes (depending on your ability with math) sounds... Unreasonable and time consuming.. Whether you can say the same about maximum and minimum is also up for debate, sure sitting down with a pencil and paper to find the minimum amount of material you have to use is time consuming. Sure, this may sound ignorant and if you got this far, I am surprised.
I'm not surprised. This sort of reasoning is very common among people who are very stressed out and finding their work difficult.

There is a person in the office here where I work whose job is to develop an algorithm that takes a computer simulation of cities and allows the calculation of the optimal use of energy in the city. That is basically a minimization problem of the energy consumption of all of the various elements of the city. It is a very difficult and non-trivial problem because many of those city elements are interacting with each other, so although it is a giant "differentiate and set to zero" problem, it is not simple at all.

I'm curious how this applies to the real world. Don't get me wrong, I'm a math major in computer science, not a pure mathematics major. I have been through a lot of calculus and every level of it with A+ but I can't seem to connect it to the real world as well; however I do see the usefulness sometimes. Any thoughts? Am I wrong about something? Please I'm open to anything. This isn't to sound like a ignorant fool. I am curious. A lot of calculus is useful to me, I loved to using it in physics and that was a very useful part of calculus: physics. But some things like solids of integration seem trivial but it's still cool to know.
I'll leave it up to you to find your own situations where calculus is useful (there are so many that a quick google search will suffice). However, let me put it this way... I'm a software developer with a back ground in physics. Although I don't use calculus on a daily basis, quite often problems involving calculus get put on my desk. If I didn't know calculus, I couldn't do my job properly and I wouldn't have even been hired.