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Lengths, areas and volumes

Lengths, areas and volumes
It’s a very classical issue in mathematics to calculate lengths and areas and volumes of geometric objects. And of course it fits right into our theme of analytic geometry, the connection between numbers and geometrical objects. Let’s begin with a circle. There’s nothing more classical than a circle. Do we remember the formula for the length of a circle or the circumference of a circle, perhaps also called the perimeter of a circle? Well if the circle has radius r, than its circumference is 2 pi r. That’s actually the definition, in fact, of the number pi, the ratio between the circumference of the circle and its diameter, the diameter meaning, of course, twice the radius.
Now suppose you cut out from the pizza a certain piece corresponding to an angle theta. What is the length along the circumference, the arc length that you get corresponding from that piece, which is called a sector in geometry? Answer– the arc length along the circumference corresponding to the angle theta is given by r theta, provided theta is measured in radians. That’s easy to see on a proportionality basis. 2 pi gives you full circle, 2 pi r circumference. Theta gives you theta r of arc length along the circumference. Now one can ask, what is the area of the circle?
You often hear that, but it’s actually not a very good thing to ask because a circle is a one dimensional object, doesn’t really have an area. To be more precise if you want to be picky about it, you’re talking about the area of a disc. The disc is what’s inside the circle. So what is the area of a disc of radius r? Answer– pi r squared. I’m sure you know that formula. There’s an old math joke about it. Warning, math joke warning. When someone says pi r squared, no, pi r round. Get it? Never mind.
By the way, if you look at ellipses, canonically ellipse, x squared over a squared plus y squared over b squared equals 1, turns out the area inside the ellipse is pi a b. That’s a generalization of the preceding formula for circles. Speaking of areas, you know that we looked at how Archimedes proved that the area of the domain between 0 and b and under the graph of y equals x squared is given by b cubed over 3. What about the length of the upper boundary of that domain D, the piece of the parabola that defines the upper boundary? Well, you can actually calculate that length. And it turns out to be this. Now Archimedes could not have found this answer.
He would have been impressed by the answer, but he would have asked, what’s that ln thing in the answer, because logarithms had not yet been defined at this point. Of course we don’t remember such a formula which is longer than the curve itself, or even the preceding one, because in calculus we can actually, in first year calculus, calculate this answer. We don’t attempt to memorize very many formulas anymore, but there are a few that we usually try to keep in mind, but just a few. Life is too short and we have too few neurons to remember everything. Besides, we have the internet, right?
Now when you look at three dimensional objects, there are considerations both of volume and also of surface area. Most classic example, a sphere of radius r, it has a surface area. Do you know the formula for the surface area of a sphere? It’s given by 4 pi r squared. How about the content of the sphere, the volume inside? Is given by four thirds pi r cubed. Did Archimedes know these formulas? Absolutely– he proved them by the same exhaustion method that we’ve seen previously. Let’s look at the surface area of two types of geometrical objects we often meet, cones and cylinders.
If you look at a right circular cone of radius r and height h– it’s a circular cone because the base, as you can see, is a circle– then there’s a formula for its surface area. First we have the base. The base, of course, is a circle of radius r. So its surface area is pi r squared. But then we have the lateral surface area of the cone. That’s given by the formula pi r square root of h squared plus r squared. If we want the total surface area, then we add these two up, of course. Or how about a cylinder? Here is a right circular cylinder of radius r and height h. We’d like to know its surface area.
Well as regards its base, once again that’s going to be pi r squared. How about the lateral surface area? The lateral surface area is given by 2 pi r h. You notice that 2 pi r is the perimeter of the base here, which is a circle, and the h is the vertical height. That’s a general fact about various kinds of cylinders, not just circular cylinders. If you take the perimeter of the base and multiply by the vertical height, you get the surface area of the lateral of the side. Now the total surface area of our cylinder? Well it will be twice that of the base plus the lateral surface area. Why twice?
Because our cylinder has a top and a bottom. And volumes of cylinders and cones, to finish on– well, consider our right circular cylinder. What is its volume? Answer– pi r squared h. You’ll notice that pi r squared is the area of the base this time and h is still the vertical height. That’s a general fact about cylinders. If you know the area of the base, whatever it might be, even if it’s not circular, and if you multiply by the vertical height h, you’ll get the volume. As for the cone, the volume of a right circular cone of radius r and height h is given by the formula one third pi r squared h.
Now do you need any longer to memorize these formulas? I’m not sure, because as you know, you can go on the internet and find anything. Archimedes, by the way, would have been very impressed by the internet I’m sure.

In this video, Francis deals with the calculation of certain lengths, areas, and volumes: a classical issue in mathematics!

You can access a copy of the slides used in the video in the PDF file at the bottom of this step.

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