Consider the function f(x)=x3+1 on the interval [0,3]. The interval is partitioned into six subintervals of equal length Δx=0.5. For each subinterval, a rectangle is constructed whose height is given by the value of f at the point halfway between the subinterval’s endpoints.
(a) Write an explicit six–term sum that approximates ∫03f(x)dx using the rectangles described, but do not evaluate numerically in this part.
(b) Evaluate the sum from part (a) to three decimal places.
(c) Without computing the exact integral, state whether the approximation from part (b) is likely to be an overestimate or an underestimate of ∫03f(x)dx. Justify your answer by analyzing the monotonicity and concavity of f on [0,3].