Let f (x) = (x – 4)2 + 9 and let Q be the region between the graph of f and the x-axis on [0,4].
Does a midpoint sum (with one sub-interval) provide an overestimate or underestimate of the area of Q?
We draw the midpoint sum.
We draw the tangent line to the function at x = 2 and chop off a corner of the rectangle.
We get a new shape that covers the same area as the midpoint sum, and doesn't cover all the of Q:
We conclude that the midpoint sum gives an underestimate of the area of Q.
Make it rain.