ONE MORE STEP TOWARD BETTER TOMORROW : Properties of the Definite Integral

Wednesday, 27 March 2013

Properties of the Definite Integral

The following properties are easy to check:
Theorem. If f (x) and g(x) are defined and continuous on [a, b], except maybe at a finite number of points, then we have the following linearity principle for the integral:

(i): $\displaystyle \int_{a}^{b}$ $\displaystyle \Big($ f (x) + g(x) $\displaystyle \Big)$ dx = $\displaystyle \int_{a}^{b}$ f (x) dx + $\displaystyle \int_{a}^{b}$ g(x) dx;
(ii): $\displaystyle \int_{a}^{b}$ $\displaystyle \alpha$ f (x) dx = $\displaystyle \alpha$ $\displaystyle \int_{a}^{b}$ f (x) dx, for any arbitrary number $\alpha$ .

The next results are very useful in many problems.
Theorem. If f (x) is defined and continuous on [a, b], except maybe at a finite number of points, then we have

(i): $\displaystyle \int_{c}^{c}$ f (x) dx = 0;
(ii): $\displaystyle \int_{a}^{b}$ f (x) dx = $\displaystyle \int_{a}^{c}$ f (x) dx + $\displaystyle \int_{c}^{b}$ f (x) dx;
(iii): $\displaystyle \int_{b}^{a}$ f (x) dx = - $\displaystyle \int_{a}^{b}$ f (x) dx;

for any arbitrary numbers a and b, and any c $\in$ [a, b].The property (ii) can be easily illustrated by the following picture:

Remark. It is easy to see from the definition of lower and upper sums that if f (x) is positive then $\int_{a}^{b}$ f (x) dx $\geq$ 0. This implies the following

If f (x) $\leq$ g(x) for x $\in$ [a, b] $\displaystyle \Rightarrow$ $\displaystyle \int_{a}^{b}$ f (x) dx $\displaystyle \leq$ $\displaystyle \int_{a}^{b}$ g(x) dx .

Example. We have

$\displaystyle \int_{0}^{1}$ (x² - 2x)dx = $\displaystyle \int_{0}^{1}$ x² dx - 2 $\displaystyle \int_{0}^{1}$ x dx .

We have seen previously that

$\displaystyle \int_{0}^{1}$ x² dx = $\displaystyle {\textstyle\frac{1}{3}}$ and $\displaystyle \int_{0}^{1}$ x dx = $\displaystyle {\textstyle\frac{1}{2}}$ ^.

Hence

$\displaystyle \int_{0}^{1}$ (x² - 2x)dx = $\displaystyle {\textstyle\frac{1}{3}}$ - 2 $\displaystyle {\textstyle\frac{1}{2}}$ = - $\displaystyle {\textstyle\frac{2}{3}}$ ^.

Wednesday, 27 March 2013

Properties of the Definite Integral

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