# How can I visualize differential equations and Integration in real life?

How can we understand differential equations and Integration in real life so that we can understand calculus easily. All we do here, at university level is memorize calculus and get the answer. We cannot relate these beautiful equations to other physical phenomenon because we just memorize and didn't understand.

• Look at the 'Active Calculus' book. It is available at activecalculus.org/single. Dec 29, 2021 at 7:07
• Just checking, do you specifically mean differential equations or just differentiation?
– J W
Dec 29, 2021 at 8:09
• Differential Equations! Dec 29, 2021 at 10:03
• – J W
Dec 29, 2021 at 12:59
• If you downvoted, please let OP know why, so he can learn to write better questions. @IbrahimOmer, today I realized that my answer might not address your question as well as you'd like. For me, your question is a bit vague. Also, my answer used integration to answer another math problem, rather than a physics problem (etc). Please let me know whether it was useful to you, and if not, what you're looking for, more specifically. Dec 29, 2021 at 17:38

You have asked two very different questions. I'll leave the differential equations for someone else. There is one particular application of integration which is my favorite last problem to do in Calc I. (We got behind this semester, and I was very sad not to have time for this. It feels like a perfect grand finale to me.)

You probably learned the formula for volumes of cones before high school. It makes sense that the base ($$πr^2$$) and the height (h) would be involved. And that it would be smaller than a cylinder. But why $$\frac{1}{3}$$? There is no easy way to see where the $$\frac{1}{3}$$ comes from (that I know of) before you can do integration.

Finding Volume of a Cone

Let's say our cone has radius R and height H. Put the point of the cone at the origin, and run its axis along the x-axis, so that its height is an x-coordinate. Then you know that it will go through the point (H,R). The line through that point and the origin is $$y = \frac{R}{H}x$$.

Now we imagine rotating that line around the x-axis, to create the cone. We also imagine slicing the cone vertically, so that the volume is made up of an infinite number of circular disks that are infinitely thin. Each disk has volume $$πr^2h$$, where r becomes $$\frac{R}{H}x$$ and h becomes dx (representing an infinitely thin bit of the x-axis), so each slice has volume $$π(\frac{R}{H}x)^2dx$$. Now we add them up. But that is exactly what the definite integral means, adding up an infinite number of infinitely small function values:

$$\int_{0}^{H}\pi \left ( \frac{R}{H}x \right )^{2}dx$$

Now the fundamental theorem of calculus tells you that you can find a value for this by using anti-derivatives: $$\int_{0}^{H}\pi \left ( \frac{R}{H}x \right )^{2}dx = \pi \int_{0}^{H} \frac{R^2}{H^2}x^2 dx = \pi \frac{R^2}{H^2} \frac{x^3}{3} \Big|_0^H =\pi \frac{R^2}{H^2} \frac{H^3}{3} =\frac{\pi}{3} R^2H$$

If that makes sense, you can also find the volume of a sphere. It's just a little bit harder.

• (Note1: This is my first time using latex so much in an answer here. It took me some serious work! Note2: If you have objections to my "infinitely small" terminology, you're welcome to suggest edits, but please make sure that they keep my explanation just as clear.) Dec 29, 2021 at 8:00
• Excellent explanation, but I think you got it wrong, 'H' should be in numerator rather than in denominator. Dec 31, 2021 at 7:21
• The $1/3$ can ultimately be explained by breaking a triangular prism into 3 pyramids of equal volume (this is how Euclid does it in Book 12 Proposition 7 of Elements). To go further and get the volume of a cone requires, well, calculus type stuff like increasing the number of sides of the prism and something like Cavalieri's principle (Euclid does this too) so it's not like it's especially easy, but at least it explains the $1/3$. Dec 31, 2021 at 14:19
• @IbrahimOmer, if you think it's wrong, please explain your thinking. Dec 31, 2021 at 19:12
• I fixed it; the final answer had an H in the denominator when it belonged in the numerator. Dec 31, 2021 at 22:42

Differentiation and integration considered by all scientists throughout the ages as one of the best sciences that guided the mind of man over all times

The fields of the use of calculus are very wide. It enters into many fields and are not limited to specific people or to those who use it only. But to almost all human beings. Here are some examples of its benefits:

1-What do we do if we are asked to calculate the amount of water required to fill a large swimming pool?

The answer is to determine the shape of the swimming pool and find its size. Therefore, we find the size of the water that will fill it. If it is a cubic or parallel rectangle, or .. or .., finding its size is not difficult in any way because these geometric shapes are regular.any student can find their size..But ... what if the shape of the swimming pool is not a regular geometric shape !! it begins with a slight gradient and then the slope descends steeply. Then the sides of the pool become curved, or semi-elliptical. Then it tends to rise slightly. Is it (easy) to find the size of the water to fill this The pool ? Of course yes !! it is the science of calculus!

2-Cars:

A car doesnt leave from the factory without knowing where the center of its mass and weight and central axis, to determine the factors of security and safety on different roads and different speeds of the car ..

This is done only by calculus.

3-Design:

The graphic engineer uses calculus to determine the difference and change of three- dimensional models and how it will change when exposed to multiple conditions. This helps him to create a very realistic environment in 3D movies or video games.In games like the Need 4 speed or GTA, all the statistics that the player sees during the race on the screen include the speed of the car and the distance between him and the contestants and the time between his car and each car of the participants in the race in the second and tenth of the second !! these calculations of differentiation and integration are done in the moment and moment while enjoying the game !! This principle applies to many games ..

4- Space

Space engineers often use calculus when planning long missions to launch a probe because they need different speeds in the probe's orbit proportional to gravity and altitude. Calculus helps them to determine all these variables with infinite accuracy !!

also uses calculus physicists, doctors, biologists, chemists, astronauts, economists, technologists and others ...

Let me pass you a quote given by a great Physicist, Engineer, Inventor.

The day science begins to study non-physical phenomena, it will make more progress in one decade than in all the previous centuries of its existence.

• Nikola Tesla.

Basically calculus is used in every single thing in our day and knowing it saves hundreds of years

I hope I helped a little .. HAVE A GREAT DAY

• "the mind of man" I believe you mean the minds of people? Dec 31, 2021 at 19:14

Ibrahim:

I would not think of it as either/or. Really, you should be able to understand and use the concepts in various ways. Algebraic symbol manipulation, "word problems" (to include not just physics, but chem, econ, business, bio, etc.), as well as graphical views. Perhaps even other conceptions (e.g. 500+ years ago, algebra was less developed and even basic algebra problems were "word problems" (but not practical situations, just verbose description of equations).

All of this said, it's very important not to exclude the symbol-pushing in a desire to get graphical or engineering approaches. Symbol-pushing was developed because it is easy and efficient and fast. Now, if you choose to add some graphical methods or practical word problems, fine. But still, make sure the kids are strong on the basic x-pushing around stuff. For one thing, many of these problems are solved by translation to/from the abstracted symbol world. Also, really the abstracted symbols are usually easier.

For Integration:

All of this said, my recommendation would be to add moderate amounts of simpler word problems at the end of lessons or chapters, once the students are practiced in the techniques. Of course some topics are more prone to this than others (e.g. related rates). Also some topics are inherently more geometrical (e.g. volumes of rotated surfaces). Or even just curve visualization (the whole blabla about finding extremum, roots, aspymptotes, etc.) But don't feel the need to make every topic have a geometric or word-problem approach. For instance, partial fractions are just a storm of algebra pushing symbols...roll with it and just treat it as what it is (a grunch of algebra). But of course Archimedes method is a very geometric topic (at least in its explication). So...be eclectic.

Also, one practical uses of integration are just "box counting" on graph paper (estimating the integral). For instance, King Hubbert made the point that with a fixed reserve of oil to deplete, the exact shape, timing, or even height of a peak (or multiple ones) in the production curve is irrelevant to the idea that in the end, the total integral under the curve (boxes on graph paper) is fixed. [His ideas were flawed as his estimates were low, plus total resource is actually affected by future prices and technology...but the basic explanation in his 1956 paper is a beautiful description of the concept of integral calculus, even for a non-ideal functional curve.]

Another knuckle-cruncher aspect of integration that any older scientist will be familiar with is weighing cut paper curves (e.g. NMR spectra) to get the area under the curve. I probably wouldn't waste time in math class on that, let the chemistry teachers do it. But if you want it, there's an activity.

For Differentiation:

Pedagogically, I love the approach that most ODE texts use to do most explication and practice FIRST in symbol pushing world and then later do some engineering type problems. Just think this is easier on the students. Word problems are hard, so it's just easier to get a new technique without a bunch of spinach about flow rates in and out of tanks or the like. But then afterwards, you can do a host of problems to at least get some exposure and intuition about differential equations in the dynamics of kinematics, fluids, electronics, radioactivity, etc. [And most STEM will get more in their physics and engineering classes, but this is a good start for them.]

Two common topics in ODE texts that have some graphical aspects are phase plane and predator prey system of equations. So if you want that aspect, well enjoy the time on those topics at least. Personally, I felt like they were a bit extraneous to the hard core symbol pushing, like in the second order non-homo constant coefficient DE. But I guess you could try to push students like me to spend time on them. (But it will be tough, since most schools don't give adequate time to cover a whole ODE course and you may be under pressure to cut topics, not do enrichment.)