GATE Mechanical 2020 Set 1 | GA | Question: 5

Question

Define $[x]$ as the greatest integer less than or equal to $x$, for each $x\in \left (- \infty, \infty \right ).$ If $y = [x]$, then area under $y$ for $x\in \left [ 1,4 \right ]$ is _______.

• A. $1$
• B. $3$
• C. $4$
• D. $6$

Comments

Answer here!   

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I used integration like [1,2), [2,3) , [3,4]

also, floor(x) = x 

first integration you get 1, second 2 and 3rd as 3 

sum up all to get [1,4] 

sum = 6

Answer 1

The graph of  $y = [x]$ for $x \in [1, 4]$.

The required area $ = (1 \times 1) + (1 \times 2) + (1 \times 3)  = 1 + 2 + 3 = 6 \;\text{unit}^{2}. \quad [\because $ Area of rectangle =  Base $ \times $ Height $]$

So, the correct answer is $(D).$

Comments

@Lakshman Patel RJIT @Kabir5454 

can you elaborate this point with an example:

[x] as the greatest integer less than or equal to x.

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y=f(x) ,

Then if x=6.6 then y=6 greatest integer less than or equal to 6.6 is 6.

Similarly x=6.99999999 then also y=6.

If x=6 then y=6

You can conclude from the graph as well. It form a steps like figure.

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great explanation bro,

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why answer is not 15,