question_answer

The area enclosed by the curve \[[x+3y]=[x-2]\] when \[3\le x<4\] is (where \[[\centerdot ]\] represents the greatest integer function)

A) \[1/2\]                   

B)        \[1/3\]                   

C) \[1/4\]                   

D)       1

Correct Answer: B

Solution :

We have \[[x+3y]=[x]-2,3\le x<4\] \[\Rightarrow \,\,\,\,\,[x+3y]=3-2=1\] \[\Rightarrow \,\,\,\,1\le x+3y<2\] So, we have to find the area bounded by lines \[x+3y=1,\text{ }x+3y=2,\text{ }x=3\]and\[x=4\] Required area \[=\int\limits_{3}^{4}{\left( \frac{2-x}{3}-\frac{1-x}{3} \right)dx=\int\limits_{3}^{4}{\frac{1}{3}dx=\frac{1}{3}}}\]