Owl: You want the area under a smooth hill-shaped curve y = f(x) from x = 1 to x = 5, but you refuse to invent a fancy formula yet. Instead you draw four clumsy rectangles that roughly cover the region.
Ugly? Yes. Useful? Extremely — because as the rectangles get thinner, the stacked areas usually settle toward a single number we will later call the definite integral.
On the whiteboard, imagine partitioning [1, 5] into four equal pieces. Left endpoints, right endpoints, or midpoints each give a different stack.
Which of those three sketches would you trust more for a first estimate — and why might "more rectangles" still not be exact?