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?

Answer

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.