# Inequalities – Property III

## Inequalities – Property III

Property: $$\left| \int\limits_{a}^{b}{f\left( x \right).dx} \right|\le \int\limits_{a}^{b}{|f\left( x \right)|.dx}$$

Proof: $$-\left| f\left( x \right) \right|\le f\left( x \right)\le \left| f\left( x \right) \right|\ \ \forall \ x\in \left[ a,b \right]$$,

$$\int\limits_{a}^{b}{-\left| f\left( x \right) \right|}.dx\le \int\limits_{a}^{b}{f\left( x \right)}.dx\le \int\limits_{a}^{b}{|f\left( x \right)|}.dx$$,

$$-\int\limits_{a}^{b}{\left| f\left( x \right) \right|}.dx\le \int\limits_{a}^{b}{f\left( x \right)}.dx\le \int\limits_{a}^{b}{|f\left( x \right)|}.dx$$,

$$\left| -\int\limits_{a}^{b}{f\left( x \right)}.dx \right|\le \int\limits_{a}^{b}{|f\left( x \right)|}.dx$$,

$$\left| \int\limits_{a}^{b}{f\left( x \right)}.dx \right|\le \int\limits_{a}^{b}{|f\left( x \right)|}.dx$$,

Example: Estimate the absolute value of the integral $$\int\limits_{10}^{19}{\frac{\sin x}{1+{{x}^{8}}}}.dx$$.

Solution:

$$|\sin x|\le 1\ \ for\ \ x\ge 10,\ \ then\ \ |\frac{\sin x}{1+{{x}^{8}}}|\le \frac{1}{|1+{{x}^{8}}|}$$…(i)

But $$10\le x\le 19\ \ or\ \ 1+{{x}^{8}}>{{x}^{8}}\ge {{10}^{8}}$$,

$$\frac{1}{1+{{x}^{8}}}<\frac{1}{{{x}^{8}}}\le \frac{1}{{{10}^{8}}}$$,

$$\frac{1}{|1+{{x}^{8}}|}\le \frac{1}{{{10}^{8}}}$$…(ii)

From equation (i) and (ii)

$$\left| \frac{\sin x}{1+{{x}^{8}}} \right|\le {{10}^{-8}}$$,

$$\int\limits_{10}^{19}{\frac{\sin x}{1+{{x}^{8}}}.dx\le \left( 19-10 \right)}\times {{10}^{-8}}$$,

$$=9\times {{10}^{-8}}=\left( 10-1 \right)\times {{10}^{-8}}$$,

$${{10}^{-7}}-{{10}^{-8}}<{{10}^{-7}}$$,

$$\left| \int\limits_{10}^{19}{\frac{\sin x}{1+{{x}^{8}}}}.dx \right|<{{10}^{-7}}$$,

Therefore, the approximate value of the integral = $${{10}^{-8}}$$.