Calculus is often portrayed as the covert king of maths; the subject is almost completely shrouded in mystery and passed off as 'too difficult' until you turn 15 or 16 years old, it's sometimes taught as an entire subject by itself rather than with other forms of maths such as mechanics or statistics, and students are often heard complaining about it continuously. Despite the massive reputation the subject boasts, calculus is based off of an unbelievably simple concept: use small changes in continuous functions to find the area underneath curves or to find the gradient of a curve.
While calculus, as an area of study, has existed since the 17th century, a relatively recent development in the field has a lot of potential to change the way we think about calculus. This development, known as visual calculus, can be used to solve problems visually which would otherwise be solved using difficult integration questions. It uses the basic concept behind calculus (small changes in continuity), and applies it geometrically rather than using calculations.
The easiest introduction to visual calculus is to find the area of a ring, or annulus, given only the length of a chord of the larger circle which is also tangent to the inner circle. This is done relatively simply using geometry, as shown by the animation below.
There are two interesting byproducts of answering this question. The first is that the area of the annulus is independent of the radii of the inner and outer circle, and is dependent solely on the length of the chord. The second byproduct is the question of why the annulus has the same area as a circle. Although a mathematical proof is given in the video above, a visual proof often makes more sense and drives the idea home much more clearly. Visual calculus helps give a more intuitive and interesting solution to the latter question. Firstly, visual calculus uses an idea called a tangent sweep; we use a tangent lines to 'sweep' out the area which we want to find. This sweep is shown in the animation below.
Now, we can use small changes in the tangents to approximate the area of the annulus more accurately. This is done by drawing a right angled triangle, with one length as the radius and one length as the small change in the tangent, which I labelled as h in the animation below. We can then move the radius to the position at the hypotenuse and repeat this process to find a many triangles which approximate the circle's area. What's more important is that we decrease the value of h to make the approximation more and more accurate. For those of you with some calculus knowledge under your belt, this is analogous to taking the limit as h approaches 0. Doing so means that any inaccuracy in the area is essentially eliminated. For practical purposes, I only used a minimum h of about 0.1, but the area of all of the triangles in the final run is almost indistinguishable from the actual area of the annulus.
Finally, we use the idea that moving around the triangles doesn't change their overall area. Therefore, moving the triangles so that they are clustered together will give us another shape which has the same area as the annulus. In other words, the area of the tangent sweep is equal to the area of the tangents clustered. This is the final key idea of visual calculus, and it is shown below. As the triangles cluster together, they form a more and more accurate representation of the overall area of the annulus, which we can see represents the area of a circle with the same radius as the chord length it comes from. This is where the circle with a radius of 4 inches truly comes from, and visual calculus gives us a quick and easy intuition for why this is correct.
Although this might seem like a more difficult way of justifying something which can be proven using 5 lines of algebra, it gives a useful intuition for other, more difficult questions. For example, visual calculus allows us to find the area under a cycloid relatively quickly, compared to the traditional method of using calculus to find the area under the curve which is defined parametrically. Moreover, since this method was only developed in 1959, there may be countless applications for it which have yet to be discovered. If nothing else, it may save you having to run through a long, tedious integration question in an exam if you could draw a few pictures and visualise the tangent sweep coming together and forming a shape in the tangent cluster.
As always, there is always more reading to be done and more to learn about this subject. Further reading links are given below, and learning contemporary calculus before learning visual calculus is highly recommended. However, that's not to say you can't explore and find out other elegant proofs using visual calculus, and I highly encourage you to do so (once you have a firm grasp of the basics).
Further Reading:
Husch, L. S., 2001. Visual Calculus. [Online] Available at: http://archives.math.utk.edu/visual.calculus/ [Accessed 7 December 2019].
Mnatsakanian, M., 2000. Visual Calculus by Mamikon. [Online] Available at: http://www.cco.caltech.edu/~mamikon/calculus.html [Accessed 7 December 2019].
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