## 2017 2/3rds numbers game

This is the fourth year that I’ve played the 2/3rds numbers game with my first year maths class. I’m always interested to see how, knowing previous results will affect this year’s results. Of course I am sure that a great deal depends on exactly how I explain the game, and so I imagine that this is the largest confounding factor in this ‘study’.
If you don’t know about the 2/3rds numbers game, take a look at the post here.

Here are the histograms from the last three years:

This year I told the class the mean results from the previous years to see if it would make a difference (as it seemed to last year). This year, the results are somewhat lower:

The winner was thus the person who got closest to 2/3 of 24.4=16.3. This year one person guessed 16, and one person guessed 16.2. Because everyone was asked to write down an integer, unfortunately I can’t claim that 16.2 is the winner, but they will get a second prize.…

## Some more volume visualisations

Here is an animation which may help you imaging a shape which has a circular base, with parallel slices perpendicular to the base being equilateral triangles:

The same thing, where the slices are squares.

And here is the region in the (x,y) plane between $y=\sqrt{x}$, the x-axis and the line x=1. rotated about the y-axis. Here a thin shell is drawn in the volume, then pulled out. Then it is replaced, then the volume is filled with shells, and each of them is pulled out of the volume vertically. This is to give you an idea about how to visualise the method of cylindrical shells.

 How clear is this post?
Gallery

## 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.

## Using integration to calculate the volume of a solid with a known cross-sectional area.

Hi there again, I have not written a post in while, here goes my second post.

I would like us to discuss one of the important applications of integration. We have seen how integration can be used to solve the area problem, in this post we are going to see how we can use a similar idea to solve the volume problem. I suggest that we start by looking at the solids whose volume we know very well. You should be able to calculate the volumes of the cylinders below (yes,  they are all cylinders.)

Cylinders are nice, we only need to multiply the cross-sectional area by the height/length to find the volume. This is because they have two identical flat ends and the same cross-section from one end to the other. Unfortunately, not all the solid figures that we come across everyday are cylinders. The figures below are not cylinders.…