Math 106-01: Calculus II: Reading Homework 7.5

1. Approximating Definite Integrals : why might we approximate definite integrals in the first place? why use some other method than left- and right-hand sums?
   
   1. the Midpoint Rule : how is the height of the rectangles in the midpoint rule calculated? how does its accuracy compare with that of left- and right-hand sums?
      
   2. the Trapezoid Rule : how is the trapezoid rule approximation calculated? what does it correspond to adding up?
      
   3. Errors : what characteristic of a function determines whether the left- or right-hand sum is an over- or underestimate? what characteristic determines this for the trapezoid or midpoint rules?