Question

The values of the following definite integrals in the decreasing order is
A) ${\int }_0^3\left[x\right]dx$
B) ${\int }_{-2}^2{x}^{3}dx$
C) ${\int }_0^3(x-[x\left]\right)dx$
D) ${\int }_{-1}^1|x|dx$

• A. A,C,D,B
• B. A,D,C,B
• C. D,A,B,C
• D. A,C,B,D

Answer 1

The correct option is A A,C,D,B

$A={\int }_0^3\left[x\right]dx={\int }_0^{1}0dx+{\int }_1^{2}1dx+{\int }_2^{3}2dx$
$=0+1+2\left(1\right)=3$
$B={\int }_{-2}^2{x}^{3}dx=\frac{x^4}{4}{]}_{-2}^2=0.$
$C={\int }_0^3(x-[x\left]\right)dx={\int }_0^{3}x-{\int }_0^3\left[x\right]=\frac{x^2}{2}{]}_0^3-3=\frac{3}{2}$
$D{\int }_{-1}^1\left|x\right|dx={\int }_{-1}^0(-x)dx+{\int }_0^{1}xdx$
$=\frac{-x^2}{2}{]}_{-1}^2+\frac{x^2}{2}{]}_0^1=[0-(+1\left)\right]+1=1$