In this problem we are given several values of the function r(t). Q(t) is simply the integral of r(t) - that is, the area under the graph. When evaluating Q(6) using the midpoint rule means:

Q(6)~Q((6-0)/2)*6.

a better evaluation can be achieved by evaluation the area between t(n) and t(n+1) using the trapezoidal rule and then summing.

a lower estimate can be achieved in several ways, two of which:

(a) using the overall minimum (2) and calculation minimum*time

(b) a more realistic lower bound can be achieved by doing (a) for every segment of the form [n,n+1] (thus using 52 as the minimum in [5,6] ) and summing.

an upper estimate is achieved similarly.