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)
Answer
(f(12) – f(0))(12) = 1728
(f(12) – f(0))(6) = 864
(f(12) – f(0))(4) = 576
(f(12) – f(0))(3) = 432
(f(12) – f(0))(1) = 144
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?
Answer
Since a, b, f(a), and f(b) are the same as in the example, we get the inequality
We solve this inequality for n:
We need to use at least 173 rectangles.
Remember that if LHS(n) and RHS(n) are close to each other, they must also be close to the exact area between f and the x-axis on [a, b].
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?
Answer
We have
Plugging this into the inequality, we get
We solve the inequality and get:
We must have more than 576 rectangles.
Example 4
Let f(x) = x4on the interval [-2, -1]. For what values of n are LHS(n) and RHS(n) within 0.5 of each other on this interval?
Answer
We have
Plugging this into the inequality, we get
We solve the inequality and get:
We must have n = 31 or greater.
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?
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?
