MAT 126Solutions to First Midterm, lemur version1. Letf(x) =12x2+12,2≤x≤6.(In what follows you are not asked to compute the exact value of a definite integral.)(a)20 pointsUse the left endpoint rule and four intervals to estimate the area of the regionunder the graph of the function.(b)10 pointsIs the estimate you have obtained an overestimate or an underestimate?(c)10 pointsIf you apply the right endpoint rule to the same function, interval and numberof intervals, do you obtain an overestimate or an underestimate?2. Consider the limit of the sumlimn→∞nXi=13n3ni3.(1)(a)20 pointsFind a functionf(x)and an interval[a, b]such that the limit above is the areaunder the graph off(x)over the interval[a, b].
Page 2 of 6(b)20 pointsUse the following formula for the sum of the firstncubesnXi=1i3=n(n+ 1)22to compute the limit of the sum in (1) (do not compute the integral directly).