Let f(x) = x2. Calculate the following quantities.
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?
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].
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?
Plugging this into the inequality, we get
We solve the inequality and get:
We must have more than 576 rectangles.
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?
We must have n = 31 or greater.
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?
We solve the inequality to get
We must have
n ≥ 6.
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