A dialogue where students discuss area approximations.

Check out this dialogue between two calculus students (based on a true story):
Devyn
Hey Riley, I’ve found a problem with approximating areas!
Riley
Fascinating! Tell me more.
Devyn
Remember how it works. We split the region into some number of rectangles, find the area of each of the rectangles, then add them all together.
Riley
Yes, and by taking more rectangles we can make that approximation better and better!
Devyn
Pretty much. But think about the calculations we’ve been doing. For each rectangle, we’re tracking the height and area, then adding them all together. That works for, like, 6 or 8 rectangles, but we want LOTS of rectangles to get a really good approximation.
Riley
How can we track this for 100 rectangles?!?!?
Devyn
Exactly!
Riley
Hmmm...
Which of the following quantities does the accuracy of our approximation depend upon?
The function. The interval. The number of rectangles.