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)

$$\int_{a}^{b}$$$$\Big($$f (x) + g(x)$$\Big)$$dx = $$\int_{a}^{b}$$f (x) dx + $$\int_{a}^{b}$$g(x) dx;

(ii)

$$\int_{a}^{b}$$$$\alpha$$f (x) dx = $$\alpha$$$$\int_{a}^{b}$$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)

$$\int_{c}^{c}$$f (x) dx = 0;

(ii)

$$\int_{a}^{b}$$f (x) dx = $$\int_{a}^{c}$$f (x) dx + $$\int_{c}^{b}$$f (x) dx;

(iii)

$$\int_{b}^{a}$$f (x) dx = - $$\int_{a}^{b}$$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]     $$\Rightarrow$$    $$\int_{a}^{b}$$f (x) dx $$\leq$$ $$\int_{a}^{b}$$g(x) dx  .

Example. We have

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

We have seen previously that

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

Hence

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

Exercise 1. Given that

$$\int_{1}^{2}$$f (x)dx = 2  ,    $$\int_{1}^{4}$$f (x)dx = - 1  ,

find $$\int_{2}^{4}$$f (x)dx.

Answer.

Exercise 2. Let f (x) be defined and continuous on [a, b]. Assume that f (x) is positive. Show that the function

F(x) = $$\int_{a}^{x}$$f (t)dt

is increasing on [a, b].

Answer.

Exercise 3. Let f (x) and g(x) be two functions defined and continuous on [a, b]. Show that

$$\left(\vphantom{\int_a^b f(x)g(x)dx }\right.$$$$\int_{a}^{b}$$f (x)g(x)dx$$\left.\vphantom{\int_a^b f(x)g(x)dx }\right)^{2}_{}$$ $$\leq$$ $$\int_{a}^{b}$$$$\Big($$f (x)$$\Big)^{2}_{}$$dx ^. $$\int_{a}^{b}$$$$\Big($$g(x)$$\Big)^{2}_{}$$dx  .

Answer.

For more on the Area Problem, click HERE.

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