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Monday, 4 August 2014

CHAPTER 2- Basic Properties

1
Let us consider the integral of {f_1}\left( x \right) + {f_2}\left( x \right) from x = a\,\,{\rm{to}}\,\,x = b. To evaluate the area under {f_1}\left( x \right) + {f_2}\left( x \right), we can separately evaluate the area under f_1(x) and the area under f_2(x) and add the two areas (algebraically). Thus:
\int\limits_a^b {\left( {{f_1}\left( x \right) + {f_2}\left( x \right)} \right)\,dx\,\, = } \,\,\int\limits_a^b {{f_1}\left( x \right)\,dx\,\, + } \,\,\int\limits_a^b {{f_2}\left( x \right)\,dx}
Now consider the integral of kf(x) from x = a  to x = b. To evaluate the area under kf(x), we can first evaluate the area under f(x) and then multiply it by k, that is:
\int\limits_a^b {kf\left( x \right)dx\,\, = \,\,} k\int\limits_a^b {f\left( x \right)dx}
2
Consider an odd function f(x), i.e., f\left( x \right) =  - f\left( { - x} \right). This means that the graph of f(x) is symmetric about the origin.
From the figure, it should be obvious that \int\limits_{ - a}^a {f\left( x \right)dx = 0,}  because the area on the left side and that on the right algebraically add to 0.
Similarly, if f(x) was even, i.e., f\left( x \right) = f\left( { - x} \right)
\int\limits_{ - a}^a {f\left( x \right)dx = \,\,2} \int\limits_0^a {f\left( x \right)\,\,dx}  because the graph is symmetrical about the y-axis.
If you recall the discussion in the unit on functions, a function can also be even or odd about any arbitrary point x = a. Let us suppose that f(x) is odd about x = a, i.e
f\left( x \right) =  - f\left( {2a - x} \right)
Suppose for example, that we need to calculate \int\limits_0^{2a} {f\left( x \right)dx} . It is obvious that this will be 0, since we are considering equal variation on either side of x = a, i.e. the area from x = 0 to x = a and the area from x = a to x = 2a will add algebraically to 0.
Similarly, if f(x) is even about x = a, i.e.
f\left( x \right) = f\left( {2a - x} \right)
then we have, for example
\int\limits_0^{2a} {f\left( x \right)dx = 2\,\,\int\limits_0^a {f\left( x \right)dx} }
From this discussion, you will get a general idea as to how to approach such issues regarding even/odd functions.
(3) Let us consider a function f(x) on [a,b]
We want to somehow define the “average” value that f(x) takes on the interval [a,b]. What would be an appropriate way to define such an average?
Let {f_{av}} be the average value that we are seeking. Let it be such that it is obtained at some x = c\,\, \in [a,\,\,b]
We can measure {f_{av}} by saying that the area under f(x) from x = a to x = bshould equal the area under the average value from x = a to x = b. This seems to be the only logical way to define the average (and this is how it is actually defined!). Thus
{f_{av}}\left( {b - a} \right) = \int\limits_a^b {f\left( x \right)dx}
 \Rightarrow\,\,\,\, {f_{av}} = \dfrac{1}{{b - a}}\int\limits_a^b {f\left( x \right)dx}
This value is attained for at least one c \in \left( {a,\,\,b} \right) (under the constraint that f is continuous, of course).
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