Ideas for Extension 
The following ideas for extending this topic require the full version of Autograph. 

Idea 1 – Estimating the Area under a Curve 
Download 1. Estimating.agg 


There are three different ways of estimating the area under a curve using Autograph 

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Challenge your students to calculate 
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Is the estimate of the area using rectangles an over estimate or an under estimate? Can you explain why? 
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What will happen to the estimate if we increase the number of rectangles? 
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Click on Animation Controller and experiment by changing the number of rectangles 
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When you are ready, doubleclick on the rectangles and choose Trapezium Rule. 
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Can you see how this calculates and estimate of the area? 
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Is it more or less accurate than using rectangles? Why? 
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Using the Animation Controller experiment with what happens when you change the number of strips. 
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What happens to the estimate if you move the two points to new positions? Predict, and then drag the points to new positions to find out! 
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Finally, try experimenting with Simpson’s Rule for estimating the area under the curve. Can you see how this technique works? 
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You can also doubleclick on the curve itself and experiment with different equations. 

Idea 2 – Infinite and Improper Integrals 
Download 2. Unbounded.agg 


We can also use Autograph to investigate integrals that have undefined or unbounded limits 

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Before opening the Autograph file, challenge your students to work out the following: 
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How about this? 
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Why is this? What is the difference between this and the first question? 
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Challenge your students to sketch the situation, and then open up Autograph. 
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Select the point at x = 1 and click on Animation Controller. Move the value closer to 1, adjusting the step size as you go. 
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What is happening to the size of the area? Will it reach a limit? Why? 
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When your students are happy with this, return the point back to x = 1. 
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Challenge your students to think what the answer to this will be: 
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Select the point at x = 2 and click on Animation Controller. Increase the value, adjusting the step size as you go. 
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What is happening to the size of the area? Will it reach a limit? Why? 
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When you are happy, try the same thing with: 

Idea 3 – Area between functions 
Download 3. Between Curves.agg 


Autograph can also work out the area between two functions. 

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Challenge your students to work out the area between the line and the curve. 
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How many different ways can they think to do this?
– Using integration and the area of a triangle
– Calculating two integrations and then subtracting the answer
– Can they do it using a single integration? 
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What happens if you do the subtraction the other way around? 
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Once students have reached an answer, click on the shaded area and select Text Box to display the size of the area. 
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You can double click on either of the lines to change the equation and test your students on other situations. 

Idea 4 – Volume of Revolution 
Download 4. Volume of Revolution.agg 


The way Autograph brings the concept of volume of revolution to life in 3D is quite something! 

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Challenge your students to picture the solid that will form if we rotate the shaded area 360° around the xaxis. 
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When you are ready, select Slow Plot, leftclick on the shaded area, rightclick and choose Find Volume 
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Hold down leftclick and drag the cursor around the screen to see the solid emerge. 
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Now you can challenge your students to work out the volume. 
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A good way to start is to select the point at x = 3 and drag it closer to the other point (see Handy Autograph Tip below). 
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Students should hopefully see that the solid begins to resemble a cylinder the closer it gets, and following on from this the entire volume itself can be thought of as the sum of lots of these thin cylinders. 
