Two young mathematicians discuss cutting up areas.

Check out this dialogue between two calculus students (based on a true story):
Devyn
So, Riemann sums compute approximate areas, but we have to pick sample points first.
Riley
Right.
Devyn
If we change sample points, what happens?
Riley
We get different areas.
Devyn
Is the area of the region changing? The region isn’t changing. Why does the Riemann sum change value?
If we change sample points, does the area of the region change?
Yes No
When we change sample points, does the value of the Riemann sum change?
Yes No
If the region’s area does not change, why is the Riemann sum changing?
Answer will vary.