Autograph Newsletter 8 – Calculus: Integration

I hope you enjoy the eighth Autograph Newsletter! Each jam-packed edition looks at a specific topic in mathematics and how Autograph can help engage students and enable them to understand the key concepts better.
Diagnostic Question
Free Online Autograph Activity
Ideas for Extension
Video Tutorials
Handy Autograph Tip
The topic of integration remains a mystery for lots of students. Many can understand, if introduced to it from first principles, that differentiation can be used to find the gradient of a curve. But why on earth should doing the opposite of differentiation magically find us the area underneath a curve? Once again, Autograph can help. Not only can you help students’ visualise their answers to standard textbook integration questions to check their validity, but more importantly you can dynamically introduce the process of numerical approximations to integration, which can be used to unlock the mystery of why integration works. And once you have mastered that, why not venture into the world of 3D for some volume of revolution?
Diagnostic Question
Diagnostic questions are ideal to use at the start of the lesson to enable you to get a quick and accurate picture of your students’ levels of understanding. They are designed in such a way that common misconceptions that your students may hold should steer them to one of the incorrect answers, thus allowing you to learn where the problems lie from their responses. Typically I give my class 30 seconds thinking time and then ask them to hold up their fingers: 1 for A, 2 for B, etc.
Reflection Diagnostic Question
Free Online Autograph Activity
The Trapezium Rule
What happens to the value of the area when you increase the number of strips? Is the area given an under or over estimate of the actual value? How can we change that?
The Trapezium Rule
These Autograph activities do not require the full version of Autograph to run them. You just need to install the free Autograph Player (you will be guided through how to do this), which means you can use these activities in the classroom or set them for your students to do at home.
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
Challenge your students to calculate 
Is the estimate of the area using rectangles an over estimate or an under estimate? Can you explain why?
What will happen to the estimate if we increase the number of rectangles?
Click on  Animation Controller and experiment by changing the number of rectangles
When you are ready, double-click on the rectangles and choose Trapezium Rule.
Can you see how this calculates and estimate of the area?
Is it more or less accurate than using rectangles? Why?
Using the Animation Controller  experiment with what happens when you change the number of strips.
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!
Finally, try experimenting with Simpson’s Rule for estimating the area under the curve. Can you see how this technique works?
You can also double-click 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
Before opening the Autograph file, challenge your students to work out the following: 
How about this? 
Why is this? What is the difference between this and the first question?
Challenge your students to sketch the situation, and then open up Autograph.
Select the point at x = 1 and click on  Animation Controller. Move the value closer to 1, adjusting the step size as you go.
What is happening to the size of the area? Will it reach a limit? Why?
When your students are happy with this, return the point back to x = 1.
Challenge your students to think what the answer to this will be: 
Select the point at x = 2 and click on  Animation Controller. Increase the value, adjusting the step size as you go.
What is happening to the size of the area? Will it reach a limit? Why?
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.
Challenge your students to work out the area between the line and the curve.
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?
What happens if you do the subtraction the other way around?
Once students have reached an answer, click on the shaded area and select  Text Box to display the size of the area.
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!
Challenge your students to picture the solid that will form if we rotate the shaded area 360° around the x-axis.
When you are ready, select  Slow Plot, left-click on the shaded area, right-click and choose  Find Volume
 Hold down left-click and drag the cursor around the screen to see the solid emerge.
Now you can challenge your students to work out the volume.
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).
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.
Video Tutorials
The following video takes you through, step-by-step, how you can use Autograph to investigate the area underneath a curve.
Handy Autograph Tip
The  Animation Controller is very handy for moving the position of points in 2D. But how about in 3D? Fear not!
Open the Volume of Revolution file.
Click on the point at x = 3.
Use the left arrow key on the keyboard to move the point.
Now, imagine you wanted things moving a bit quicker – do the same again, but this time hold down theShift key.
And if you want more precision – simply hold down the Ctrl key whilst hitting the arrow key.
Note: This method also works in 2D!

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