The Area Problem
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Area and Definitive Integrals
This video just gives a historical overview of the area problem. Now, in ancient times, even before recorded history, people have been able to figure out things like the area of a rectangle or the area of a triangle. And the ancient Greeks got a little better with geometry and they could do all polygons. And then the amazing genius Archimedes came along.
And with his amazing formula, area equals pi r squared, the human race knew the area of a circle. That was an extraordinary advance. Well, a couple other odd areas were figured out in antiquity, but really the whole problem how to find the area of irregular shapes stopped for several centuries.
Now, of course, part of the problem was the dark ages and not much progress was made in the dark ages. But when calculus was invented, then a whole new class of areas could be found. In particular, when we have a function f(x). And that function could be any one of a number of different kinds of functions, then we could find the area under that function.
Well, that opens up a whole new class, of kinds of areas we can find. And, indeed, using calculus, and related techniques, we can find the area of all kinds of shapes, now. And, of course, calculus was invented by Sir Isaac Newton, and independently invented, by Mr. Gottfried Leibniz. Newton, I'll just point out, is one of the most brilliant human beings ever to walk the face of the Earth.
And his mathematical contributions really overshadow Leibniz. Leibniz was a more diverse thinker. He was also a philosopher. But what we have here, actually, just in this list, these two gentlemen Archimedes and Newton, are generally considered two of the top three mathematicians of all times, the third being Mr. Leonard Euler, who has been discussed in other videos.
And so, we're going to be exploring the area problem now, using the techniques of calculus.
Read full transcriptAnd with his amazing formula, area equals pi r squared, the human race knew the area of a circle. That was an extraordinary advance. Well, a couple other odd areas were figured out in antiquity, but really the whole problem how to find the area of irregular shapes stopped for several centuries.
Now, of course, part of the problem was the dark ages and not much progress was made in the dark ages. But when calculus was invented, then a whole new class of areas could be found. In particular, when we have a function f(x). And that function could be any one of a number of different kinds of functions, then we could find the area under that function.
Well, that opens up a whole new class, of kinds of areas we can find. And, indeed, using calculus, and related techniques, we can find the area of all kinds of shapes, now. And, of course, calculus was invented by Sir Isaac Newton, and independently invented, by Mr. Gottfried Leibniz. Newton, I'll just point out, is one of the most brilliant human beings ever to walk the face of the Earth.
And his mathematical contributions really overshadow Leibniz. Leibniz was a more diverse thinker. He was also a philosopher. But what we have here, actually, just in this list, these two gentlemen Archimedes and Newton, are generally considered two of the top three mathematicians of all times, the third being Mr. Leonard Euler, who has been discussed in other videos.
And so, we're going to be exploring the area problem now, using the techniques of calculus.