Question

Answer Question . SOLVE BY USING GRAPH METHOD.​

Answer 1

AnswEr :

Option c (-π) is the answer.

ExplanaTion :

• When u find a greatest integer function, modulus function and trigonometric function in definite integral problems then solving the problem graphically makes the answer easy.

Plot the graph for | sinx | and | cosx |

REFER ATTACHMENT (1)

• We can notice that the graph is symmetric at π.

• Which means , the values from 0 - π and π - 2π are exactly same.

Given,

$\int (|sinx| - |cosx| )dx \: \: \: \: \: (0 \: to \: 2\pi)$

Now, we can reduce the graph from ( 0 - 2π ) to 2 times of ( 0 - π ).

$\implies \: 2\int (|sinx| - |cosx|)dx \: \: \: \: (0 to \: \pi)$

Now, consider only half of the graph,

Divide the domain from ( 0 - π ) into 3 parts, as the values of | sinx | and | cosx | at π/4 and 3π/4 are same.

Now, consider functions at these 3 respective domains ,

$(0 \: to \: \frac{\pi}{4} ) \: - | \sin(x) | < | \cos(x) |$

$( \frac{3\pi}{4} \: to \: \pi) - | \sin(x) | < | \cos(x) |$

$( \frac{\pi}{4} \: to \: \frac{3\pi}{4} ) - | \sin(x) | > | \cos(x) |$

But, The function of our integral lies between (-1) and (1) .

-1 < f(x) < 1.

(since, sinx and cosx lies between - and + 1)

Greatest integral function which lies in these regions :

=> It's the sum of three domains, (0-π/4) , (3π/4 - π/4) , (π - 3π/4).

1st domain ,

$2 \int f(x) \: d(x) - (0 \: to \: \frac{\pi}{4} )$

here, f(x) < 0

=> [f(x)] = (-1)

3rd domain ,

$2 \int f(x) \: d(x) - ( \frac{\pi}{4} \: to \: \frac{3\pi}{4} )$

here, also same as 1st domain,

[f(x)] = (-1)

2nd domain,

$2\int \: f(x) \: d(x) - ( \frac{3\pi}{4} \: to \: \pi)$

here, f(x) > 0

[f(x)] = 0

Now, to the simplest we got three domains ,

1st one,

$2 \int( - 1)dx \: \: \: \: (0 - \frac{\pi}{4} )$

on simplification,

$\implies \: ( \frac{ - 2\pi}{4} )$

2nd one,

$2 \int \: 0 \: dx \: \: \: \: ( \frac{\pi}{4} - \frac{3\pi}{4} )$

on simplification

$\implies(0)$

3rd one,

$2 \int \: (1) \: dx \: \: \: \: ( \frac{3\pi}{4} - \pi)$

on simplification,

$\implies \: ( - \frac{2\pi}{4} )$

Sum of three domains is the answer,

$\implies \: \frac{ - 2\pi}{4} + 0 \: + \frac{ - 2\pi}{4}$

$\implies \: ( - \pi)$

Therefore,

the answer is (-π)

option : c