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Guidelines for visualising and calculating volumes of revolution

I have seen some people try to blindly use the formulae for volumes of revolution by cylindrical cross-sections and by cylindrical shells, and I thought that I would write a guide as to how I would recommend tackling such problems, as generally just using the formulae will lead you down blind alleys.

I’ve created an example, with an animation, which I hope will help to master this technique.

So, here is a relatively fool-proof strategy:

  1. Draw the region which you are going to have to rotate around some axis. This will generally be a matter of:
    • Drawing the curves that you have been given
    • Finding where they intersect
  2. Draw the line about which you are supposed to rotate the region
  3. Draw the reflection of the region about the line of rotation: This gives you a slice through the volume that will be formed
  4. Now you have to decide which method to use:
    • Take a slice through the volume perpendicular to the axis of rotation.
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By | September 13th, 2017|Courses, First year, MAM1000, Uncategorized, Undergraduate|2 Comments
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Chaos from differential equations

In all of this talk about differential equations, we haven’t spoken all that much about the uses of them, apart from a little about population dynamics, nor indeed about their amazing properties. Part of the reason for this is that in general (though of course not exclusively), the most interesting differential equations are a single step beyond what we have been looking at. They are differential equations in more than one variable. For instance, rather than just having a y be a function of x or t, they have y a function of both x and t. It turns out that this little change makes all the difference in the world. All of a sudden we can see how things change in both space and time. We can look at real dynamics of systems which are not local to a single place.

This is a topic for another time, and comes under the term partial differential equation.…

By | September 15th, 2015|English, Level: intermediate|0 Comments
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The Fundamental Theorem of Calculus, part 1 (part i)

We’ve seen some intriguing things in this course so far, and we’ve developed some clever tricks, from how to find the gradient of just about any function we can throw at you, to proving statements to be true for an infinite number of cases.

To some extent, this is what we have looked at so far (at least in terms of calculus, and building up to calculus):

Screen Shot 2019-07-09 at 16.06.58

However, we’re about to see some magic. We’re about to see the most important thing yet on this course, and indeed one of the most important moments in all of mathematical history.

We are going to see…actually, we are going to prove, that there is a relationship between rates of change and the area under a graph. This doesn’t sound that amazing, but its consequences have essentially allowed for the development of much of modern mathematics over the last 350 years.

The link that we are going to prove will allow us to find the area under graphs of functions for which taking the Riemann sum would be really hard.…

By | July 9th, 2019|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment
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