GDC instructions for definite integrals involving areas

1. Open the area under graph applet at [https://www.desmos.com/calculator/x4f54hxohc].
2. On the sliders in the left hand pane, set \(n=1\), \(a=0\), \(b=3\).

Sketch the graph obtained and calculate the area of the shaded region by geometry.

Write down the value of \(c\) which is defined as \[\int_{a}^{b} x^{n} \, \mathrm{d}x\] What can you say about the quantities?

Test your conjecture by adjusting the slider to vary the values of \(a\) and \(b\).
For this step, keep your values of \(a\) and \(b\) bigger than \(0\).

3. On the sliders, set \(n=1\), \(a=-3\), \(b=0\).

Sketch the graph obtained in the space below and calculate the area of the shaded region by geometry.

Write down the value of \(c\) which is defined as \[\int_{a}^{b} x^{n} \, \mathrm{d}x\] What can you say about the quantities?

Test your conjecture by adjusting the slider to vary the values of \(a\) and \(b\).
For this step, keep your values of \(a\) and \(b\) smaller than zero.

4. Keep the value of \(n=1\), but adjust the values of \(n\) to be any value.
   What do you observe about the value of \(c\) vs the area measured?
5. Repeat steps \(1\) to \(4\) but with three other different values of \(n\).
   Write down your observations and learning points and share it with a classmate.
6. Look back
   What is the difference between the area under graph and \(\int_{a}^{b} \mathrm{f}(x) \, \mathrm{d}x\)?
   Why is it important to sketch the graph of \(y = \mathrm{f}(x)\) before finding the area under a graph?