Example 1
Let f(x) = x2. Calculate the following quantities.
- (f(12) – f(0))(12)
- (f(12) – f(0))(6)
- (f(12) – f(0))(4)
- (f(12) – f(0))(3)
- (f(12) – f(0))(1)
Example 2
Let f(x) = x2 on the interval [0, 12]. How many rectangles must we use to guarantee that the right- and left-hand sums are within 10 of each other?
Example 3
Let f(x) = 4x + 6 on the interval [2, 8]. How many rectangles must we use to ensure that the left-hand sum is within 0.25 of the exact area between f and the x-axis on this interval?
Example 4
Let f(x) = x4 on the interval [-2, -1]. For what values of n are LHS(n) and RHS(n) within 0.5 of each other on this interval?
Example 5
Let f (x) = cos(x) on the interval 
How large must n be to guarantee that LHS(n) is within 0.3 of the exact area between f and the x-axis on this interval?
