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


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) ≤ f(x) dx ≤ M(b − a).

f(x) dx ≤ M(b − a).

Use this property to estimate the value of the integral.

8

(10x − 20 sin(x)) dx

7

Answer 1

Use the fact that -1 ≤ sin(x) ≤ 1.

Then

20 ≥ -20 sin(x) ≥ -20 … … … (multiply each side by -20)

-20 ≤ -20 sin(x) ≤ 20 … … … (swap the endpoints)

10x - 20 ≤ 10x - 20 sin(x) ≤ 10x + 20 … … … (add 10x to each side)

and over the interval [7, 8], we have

7 ≤ x ≤ 8

70 ≤ 10x ≤ 80 … … … (multiply each side by 10)

so that the whole integrand is bounded by

70 - 20 ≤ 10x - 20 sin(x) ≤ 80 + 20

50 ≤ 10x - 20 sin(x) ≤ 100

Take m = 50 and M = 100. It follows that

$50(8-7) \le \displaystyle \int_7^8 (10x-20\sin(x)) \, dx \le 100(8-7)$

which places the value of the integral between 50 and 100.