Correct Answer - Option 4 : 1
Concept:
Greatest Integer Function: (Floor function)
The function f (x) = [x] is called the greatest integer function and means greatest integer less than or equal to x i.e [x] ≤ x.
- \(\rm \int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx\)
Calculation:
Given: \(\rm \int_{0 }^{2} [x]dx\) where [x] denotes greatest integer function
Let f(x) = [x]
As we know that f(x) = 0 when 0 < x < 1 and f(x) = 1 when 1 < x < 2.
As we know that, \(\rm \int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx\)
\(\rm \Rightarrow\int_{0 }^{2} [x]dx=\int_{0 }^{1} (0)dx+\int_{1}^{2} (1)dx\)
\(\rm \Rightarrow\int_{0 }^{1} (0)dx+\int_{1}^{2} (1)dx=\left [ x \right ]_{1}^{2}=(2-1)=1\)
Hence, the correct option is 4.
