If ‘[.]’ is greatest integer function , then value of \( \int_0^{1.5}\displaystyle [x^2]\,dx \) is equal to :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( 2-\sqrt{2} \)
The greatest integer \( [x^2] \) changes value where \( x^2 \) crosses an integer, i.e. at x = 1 and \( x=\sqrt{2} \):
\( [x^2]=0 \) for \( 0\le x\lt 1 \)
\( [x^2]=1 \) for \( 1\le x\lt\sqrt{2} \)
\( [x^2]=2 \) for \( \sqrt{2}\le x\le 1.5 \)
\( \displaystyle \int_0^{1.5}[x^2]\,dx=0+1\cdot(\sqrt{2}-1)+2\cdot(1.5-\sqrt{2}) \)
\( =\sqrt{2}-1+3-2\sqrt{2} \)
\( =2-\sqrt{2} \)