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 = \({{10}^{-8}}\).