My Mathematics tuition teacher is an excellent man. Not only does he encourage creative and extra-curricular thinking, but also when you present to him a doubt, he first himself understands it to the depth and then tries to find a multitude of examples to explain it away. Even when he has explained something, he keeps thinking about it for days to come, challenging his theory in various ways, searching for easier ways to explain.
So those days, we were doing Integrals. To be honest, I was getting slightly bored because I had been doing this for a long time. As you might have read in this article in Lockdown Mine, my exams had been postponed and I had got months of extra time, which was slightly painful because I had to go through everything once again now.
To combat the boredom, I began thinking of a problem I had got a glimpse of but vaguely remembered, from a YouTube video. There is this excellent Mathematics teacher from Australia, Eddie Woo, who also has a successful YouTube channel. The video I am talking about was this:
I will try to explain the problem in a simple manner. Say, you have got with you this shape of a boa constrictor eating an elephant:
With some knowledge of Integrals, you can find the exact area of this shape between the orange line and the x-axis. If the shape were a circle, the area would have been easier to find; were it an ellipse, it would have been almost as easy. With this kind of shapes, the process is a bit tricky, nevertheless it is possible. Enough talk to give an idea about what integration can do.
Coming to the problem I encountered, let’s take an example. Say you have this shape:
Suppose the area you have got to find is this mango-color region:
Say the equation of this weird-looking shape is w(x), w for weird. While sitting at the home of my tuition teacher, I tried finding the area of this shape through two different ways. Say the two end points of this shape are 0 and 17, and the two zeroes (a zero is a point where the curve cuts the x-axis) are, say, 8 and 11.
The first method I tried the find the area A using was this:
And the second method was:
Even if you do not understand this strange Newtonian language, and seem to get afraid by these overstretched S’es, you might have got the idea that in the first method, we are trying to put in the equation in some S-shaped machine and finding area in one go right from 0 to 17. In the second method, we are finding area by breaking the shape into three parts, because that seems more natural to do.
Surprisingly, the resultant answers that came from the two methods didn’t match. I was in a crisis: had I missed some important concept in my learning of integrals that I ought to know?
When I discussed my question with my teacher, he seemed to be confused too for a while. It seemed he had either never encountered such a problem, or that he had not at least seen it in a long while.
On the first day of our quest, he tried to explain it away, but he couldn’t verify which answer was correct and why both methods were showing different answers.
The next day when I went to him, he was ready with a notebook and a pen. He asked me, when I was finished with my curricular revision, to explain the problem once again in clear words and with clear diagrams.
The third day when I went to his house, I discovered a smiling face waiting for me already, eager to explain the solution finally. My teacher had done his homework!
He had discovered that in all such cases, one could find two sorts of areas: a Net signed area, and a Total area.
When you count the area lying in the negative y-axis, or below the x-axis, as something negative, something that cancels the positive, then what you are trying to find is the net signed area. A car started on its journey. The distance it is covering is a positive number. At some point in the journey, it is running out of gas, and the nearest gas station is behind it and not ahead. It takes a U-turn. Now the distance it travels is in the negative direction, if the end-thing you are trying to find is displacement of the car. If you are interested in the total distance travelled by the car, then there is no room for negatives. Negatives are non-existent, and distance in any direction is positive distance.
Similarly, if you want to see area under x-axis as something negative, you arrive at net-signed area. If any axis means nothing to you and all you are interested in is the shape and the area covered by it, then, my buddy, you are trying to find the Total Area.
That was the difference that had been perplexing me. It might have perplexed many scholars before my teacher and me, and some wise one must have come up with the two sweet terms.
My teacher remembered that he had studied the concept before but had never heard about the two terms.
Let’s take one easy example, neither a boa constrictor swallowing an elephant, nor w(x). It’s a simple, gullible, smooth and baldly sine curve.
Between 0 and 2π, the Net Signed Area is zero, but Total area is 4 square units.
This was a wonderful discovery, that will go with me very far, and will work as an interesting riddle I’ll ask many people with utmost curiosity.
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