Sometimes it would seem that learning mathematics is hardly worth the effort. All those painful techniques and formulas, replete with grotesque and hideous symbolism, would detract even the heartiest from diving headfirst into this strange world. Yet when you come to understand that such arcane features actually serve purpose, you begin to realize that mathematics solves very difficult problems with an economy that would make the most niggardly cheapskate proud. Such is the case with calculus. Here we look at how this discipline allows us to calculate the exact volume of some very bizarre shapes.

When one of my math professors first broached the idea to me that calculus allowed us to do such things, I truly believed that he was drinking way too much wine. To my uninitiated mind, I believed that he was referring to the attempt to calculate such volumes, not the exact accomplishment. When I studied the calculus a year or so later and learned the method, I was quite surprised by the result. At this point, I thought there was nothing that math could not solve.

In a method completely analogous to that laid out in my article Why Study Calculus? - Areas of Irregular Shapes, the technique to compute volumes of irregular shapes hinges on the simple formula for calculating the volume of a disk. The formula for the volume of a disk is given by pi*r*r*h, where pi is the famous mathematical constant, approximately equal to 3.14; and r and h are the radius and height, or thickness, of the disk, respectively.

Let us show how this method would be applied to compute the volume of the following irregular shape. Picture the right half of the parabola y = x^2 on a cartesian coordinate plane (graph paper). If the reader is not familiar with the parabola, just imagine a curved line extending from left to right, much like the inside of a bowl. When we discuss the solid figure produced below, this will become clearer.

Since this graph goes on forever, let's confine ourselves to the values for which both x and y are between 0 and 2. If we now rotate this section of the parabola around the x-axis, we will produce a solid shape-known as a solid of revolution in the calculus-which sort of resembles a megaphone.

As we did with the rectangles in computing the area of irregular shapes, we do so with disks in order to compute the volume of this "megaphone." If we divide the interval from 0 to 2 along the x-axis by quarters, we can fit 8 disks of thickness 1/4 along this interval to approximate the volume of this shape. The height or radius of each disk would be determined by the point at which the individual radii intersected the parabola above. (The best way to envision this is by drawing a picture.)

Now we know the volume of a disk. This is computed quite readily. By computing the volume of each of the 8 disks above, we would have an approximation to the volume of the megaphone; however, we are not satisfied with an approximate value. We want exactitude. The problem in using 8 disks is that some disks will fall completely within the megaphone, and some will fall outside, rendering an incomplete volume. To rectify this problem-you guessed it-we further subdivide the interval from 0 to 2 into increasingly smaller subdivisions, permitting more and more disks to fit inside the megaphone. As we divide the interval more finely, the thickness of each disk gets smaller, and consequently we can fit more and more inside, thus approaching the volume of the megaphone with more precision.

Thus if we use 100 disks, we get a good idea of the volume; a 1000, even better; a million, great, but still not perfect. Now here is where the wonderful calculus comes in. By passing to the limit, which is another way of saying fitting infinitely many disks into the megaphone, we get the exact volume of this most unusual shape. What you say? How can you add the volumes of infinitely many disks?

Good question. And here is where the calculus becomes a marvel in and of itself. By analyzing the nature of the problem, that of fitting more and more disks into this interval, we can devise a formula, using the calculus, which takes the summing of these infinitely many disks into consideration, and renders the answer without having to do the actual sum. Nonsense you say? No real sense, I say.

And such is what makes studying the calculus a joy to behold, a wonder to consider, and sometimes a headache to experience. But where else can you do such crazy things? If you really are interested go learn more. Remember. Arithmetic is the gateway to algebra, this the gateway to calculus, and this the gateway to...yes the universe!

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