Question

If m ≤ f(x) ≤ M for a ≤ x ≤ b, where m is the absolute minimum and M is the absolute maximum of f on the interval [a, b], then m(b − a) ≤ b f(x) dx a ≤ M(b − a). Use this property to estimate the value of the integral. π/9 5 tan(3x) dx π/12 (smaller value) (larger value)

Answer 1

Answer: the value of the integral is in the range:

0.006 ≤ int ≤ 0.008

Explanation:

First, we have the function

5*tan(3x), and the range in this case is: pi/12 ≤ x ≤ pi/9

Then, the first step is find the maximum and minimum of f(x) in this range.

We know that between 0 and pi/2, tan(x) is a growing function, then

then the limits are

5*tan(3*pi/9) = 5*tan(pi/3) = M= 0.091

5*tan(3*pi/12) = 5*tan(pi/4) =m = 0.069

Then the value of the integral is between:

0.069*(3.14/9 - 3.14/12) ≤ int ≤ 0.091*(3.14/9 - 3.14/12)

0.006 ≤ int ≤ 0.008