You know I thank God that I was given the ability to communicate math to the masses. I guess this talent comes from my struggles with this subject early on. My ability to convey the guts of this subject derives from my belief that if I can understand it, anyone can. Such is the case with calculus. Read on as I show you how this subject allows us to calculate the exact area of even bizarrely shaped objects.
Now you all have learned from basic geometry how to find the area of such common shapes as the square or rectangle. Heck. Some of you might even remember how to find the area of shapes like the triangle or trapezoid. But how would you find the area of an irregular shape like that of an open rectangle, the top of which is formed by some winding curvy line? In other words, picture a rectangle. Now take off the line which forms the top part of it. Draw a curvy line from the left side all the way to the right side, so that the figure is now closed in space. Feel free to make the curvy line as complicated or convoluted as you like, as long as the curve does not intersect itself. Can you picture something like this? Well good then, because these are the kinds of shapes, the areas of which, the calculus will give us with the utmost precision!
How does calculus do this. Well it all begins with approximation. As mentioned in a previous Calculus article, the two main branches of this subject are differential and integral. The branch which deals with areas of irregular shapes is the integral, and this name is what we give to the mathematical object that actually calculates the area.
We all know how to compute the area of a rectangle. Now imagine we take this larger rectangle with the curvy top, and divide it into five sections in the following manner. We mark off five points on the base of the rectangle so that each one divides the base into five equal parts. At each of these points, we draw a vertical line from the point at the base to a point which is on the curvy line at the top. From this point, we form a smaller rectangle by drawing a horizontal line from the point on the curvy line across to the left so that the width of this top is the same as the width of the base. Can you envision this?
We do this with each of the evenly spaced points at the base of the rectangle, constructing a smaller rectangle from the larger one. You may have assumed correctly that by adding the areas of the smaller rectangles we can approximate the area of the larger one. In fact this is the process which leads to the integral, that wonderful mathematical object which will give us the exact area.
Since we have a curvy line at the top of the larger rectangle, we will form five smaller rectangles which in some cases lie within the larger rectangle, and in some cases lie outside the larger rectangle. You really should try to draw this to see what is happening. At any rate, we can get closer to the exact area if we divide the base into smaller and smaller partitions. Thus rather than having five rectangles, we have one hundred, or even one thousand. To get to the integral, we use an infinite number of rectangles at which point the width of each one is zero and the height is just the height from the base of the larger rectangle to the curvy line at top. Wow, what a mouthful! Chew on that for a bit.
In essence, to get the exact area of the irregular shape, we add up an infinite number of rectangles whose width is zero. Actually, we take a limit as the width of each rectangle approaches zero and an infinite number of them. From this striking example, we come to the momentous conclusion that infinity multiplied by zero is some finite number, which in this case is the exact area of the irregular shape! Welcome to the world of calculus. Now you see why this stuff is so fascinating?
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Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Under the penname, JC Page, Joe authored Arithmetic Magic, the little classic on the ABC’s of arithmetic. Joe is also author of the charming self-help ebook, Making a Good Impression Every Time: The Secret to Instant Popularity; the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart. The diverse genre of his writings (novel, short story, essay, script, and poetry)—particularly in regard to its educational flavor— continues to captivate readers and to earn him recognition.
Joe propagates his teaching philosophy through his articles and books and is dedicated to helping educate children living in impoverished countries. Toward this end, he donates a portion of the proceeds from the sale of every ebook. For more information go to http://www.mathbyjoe.com