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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 [ab], 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 [ab], 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$ [ab].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$ [ab]     $\displaystyle \Rightarrow$    $\displaystyle \int_{a}^{b}$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$\displaystyle {\textstyle\frac{1}{2}}$ = - $\displaystyle {\textstyle\frac{2}{3}}$   .
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