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Square autograph

we're going to have a look at the equation of a torus by rotating the area of a circle about a line in three dimensions and we can never look at the principles of calculus that are involved if I select the volume and delete it you can see what we're talking about here's the area of a circle and there is a line now if we have vertical elements each element of volume will be a bit like a cylinder and we can use the constant controller to vary that we'll show you how all this works later on equally if you use the elements the other way then what you get is a kind of washer and again we can vary the value of B let's have a look the problem as it arrived in an email first we're just going to tidy up the Squared's this is a text box in autograph so which i've pasted it into I'm just going to change the rate to the power 2 into Squared's and what we want to do is to select that and copy it because that's the circle whose area is going to be rotated about the line x equals to a so we'll keep that as it is and this is the diagram I'm hoping to emulate so we'll cancel this one and show you can see it from scratch and open up a new 3d graph page and right click enter equation right click paste there's the equation that we copied off the text box now a is a parameter and it takes her to full value of 1 which happy with that so here we go ah now there's no Zed in this equation so it is double-click behaving as a cylinder but if we plot it as a 2d equation we should get something a bit more appropriate so X along here y is up here it's not coping very well at the ends so what we'll do we'll double click on this and go back to the startup options and set it to manual and 0.001 will be a good setting of this step to make sure or that it closes bit better that's better okay now to get the area between the top half and the bottom half first of all I get to press control and just drag down so we can see what we're doing a bit closer I'm going to put a point on the top half and the point on the bottom half that way you can be sure to get an area between the top and the bottom I should explain that autograph interprets this as y equals plus and y equals minus the square root of a squared minus x squared so there's the plus and there is the minus so I want to double click on this one and set its value to minus a and I want to set the value to this one two plus a so now I can be sure that one belongs to top half and this one belongs to the bottom half and we should right click now be able to find the area and I'm going to use Simpsons rule and five divisions and there we go so we've got a nice full area here we now want to put on the axis so I'm going to enter an equation of X equals to a if I click OK I actually get a plane so double click on that once again plot it as a 2d because that has no said or Y so do you select everything select the area select the line and right-click find the volume that's grayed out because we've chosen a line and round it goes now it's actually going off the screen a bit it's on slope lock which is so deliberate so we get a good feel as I just progressing so what we could do is go to the Edit axes and change the X from minus 4 to plus 4 2 minus 2 tab 6 and that will just move it along a bit to give us a bit more the scope so now I'm going to select that and delete it and now let's consider what would happen if instead of a complete arrow we just had a small strip so I'm going to change this one double-click to a new parameter B B will be one to start with but if I use the console controller I can change B to say 0.5 then I want to change this one double-click to B plus 0.1 that'll give us a small element of area and there it is now if I double click on that at the moment is still set to Simpsons rule but if I just do an ordinary rectangle and only one of them I will generally get a rectangular area so now I select that and the axis and right-click find the volume and that should give us a nice cylindrical element now the console control is in place is therefore if I change B it should take the element around the cycle and even off the edge but what we want to do now is to have the elements horizontal in order to do that we're going to have to treat this graph not as a y equals plus and y equals minus but it's an x equals plus and an x equals minus and that's what's happening on the next screen so again we'll start from scratch on this one and have a new 3d graph page and right click enter equations now we want to put in the equations the other way around so x equals plus square root of a squared minus y squared don't forget to plot us a 2d and startup options as a point at 0 0 1 that's looking good so now we do the same me again right click enter equation we can take it off the history because we want it's that one isn't it so we'll just do that and put in a - same thing but I forgot to do double click plotters the 2d and all of the startup options manual point zero zero one once again control and drag so this time I want to put a point on the top branch and the point on the bottom branch so to get the element that I'm interested in office I could do the full area like I did before I'm going to set this to a value of a parameter B I've left control down that's why it's not working there we are B and click OK now B will be set to 1 but as before we can change that back to 0.5 or there abouts doesn't matter where I want this one to be B plus 0.1 so once again I get a nice element so I select this point and this point and right click find the area I want a rectangle and I just want one of them there we go we need to put in our axis enter equation x equals to a ok double click what is a 2d but first of all we need to select the element of area and the axis to get our element of volume to find the volume that's grayed out so off we go and it sort of wash up shape and once again the value of B can be varied so you can see the full family of possibilities so that's two quite interesting ways of looking at this generation of the torus you either produce a washer shape with elements horizontally or you produce a cylindrical shape using vertical elements and that was a nice solution to a problem