Let g be a continuous function that is decreasing and concave upward on the closed interval [1,4]. The interval is divided into six subintervals of equal length. A rectangular approximation of ∫14g(x)dx is obtained by, on each subinterval, drawing a vertical segment to the curve at the point located halfway between the endpoints of that subinterval and using the resulting rectangle’s area.
(a) Determine whether the rectangular approximation described is an overestimate or an underestimate of ∫14g(x)dx. Provide a clear mathematical explanation that references the stated properties of g.
(b) Suppose instead that g is decreasing and concave downward on [1,4] while all other conditions remain the same. Would the rectangular approximation be an overestimate, an underestimate, or could it be either? Justify your conclusion.