Use this property to estimate the value of the integral.

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) ab f(x) dx ≤ M(b − a).

Use this property to estimate the value of the integral.

30 (x^3-3x+5) dx

Smaller Value =

Larger Value =

Answer 1

f(x) = x³ - 3x + 5

f'(x) = 3x² - 3 = 0 when x = 1 and -1.

m = minimum value of f(x) on [0,3] = f(1) = 3

M = maximum value of f(x) on [0,3] = f(3) = 23

So, 3(3-0) ≤ ∫_{(from 0 to 3)} f(x)dx ≤ 23(3-0)

9 ≤ ∫_{(from 0 to 3)} f(x)dx ≤ 69