Answer

Verified

393.9k+ views

**Hint:**

The relation between number ‘x’, integral part of x, ‘[x]’; and fractional part of x, ‘{x}’ is given by

$x = [x] + \{ x\} $

Any function which is in the form of min(x,y) takes a minimum value from x and y in a given interval.

**Complete step by step solution:**

Step 1: - Here is the first step,

$x = \left[ x \right] + \left\{ x \right\}$

$\left\{ x \right\} = \;x - \left[ x \right]$

$x - \left[ x \right] = \left\{ x \right\}$ …………….(i)

Similarly:

$ - x - \left[ { - x} \right] = \left( { - x} \right) - \left[ {\left( { - x} \right)} \right]$

$ = \left\{ { - x} \right\}$

$ = 1 - \left\{ x \right\} \ldots \ldots \ldots \ldots \ldots \ldots ..\left( {ii} \right)$

Step 2: - In next step, We will use the graph for draw the graphical representation of (1) and (2)

Step 3: - So, In third step, We need to Combined graphical representation of (1) and (2)

Step 4: In the last step, Here we need to find the minimum bounded region

Now $\int_{ - 2}^2 {min\{ x - [x], - x - [x]} \} $$ = 4 \times Area\;of\;\Delta ABC$

$ = 4 \times \dfrac{1}{2} \times base \times height$

$ = 4 \times \dfrac{1}{2} \times 1 \times \dfrac{1}{2}$

$ = \;4 \times \dfrac{1}{4}$

$ = 1$

$\int\limits_{ - 2}^2 {min\{ x - [x], - x - [x]\} = 1} $

**Note:**

Geometrical representation of a definite integral. Let f be a continuous function on [a, b] and f(x) >= 0 x belongs to [a, b], then $\mathop \smallint \nolimits_a^b f\left( x \right)\;dx$ represents the area of the region bounded on the left by the line x = a on the right by x = b below by x – axis and above the curve y = f(x)

Given curves represented by y= f(x), y-= g(x) where f(x) >= g(x) in [a, b]

Elementary strip has height f(x) – g(x) and width dx so that the elementary area = [f(x) – g(x)dx, and total area takes an A = ∴ $\mathop \smallint \nolimits_a^b [{\text{f}}\left( x \right) - g\left( x \right)]dx$

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

List some examples of Rabi and Kharif crops class 8 biology CBSE

Which are the Top 10 Largest Countries of the World?

The provincial president of the constituent assembly class 11 social science CBSE

Write the 6 fundamental rights of India and explain in detail