Question

If \([x]\) denotes the greatest integer function, then \[ \int_{0}^{5}[x-2]\,dx= \]

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(3\)

Hint:

For greatest integer function problems, always divide the interval at the integer points where the function changes its value, and evaluate the integral piecewise.

Answer 1

The Correct Option is A

Step 1: Split the interval according to the greatest integer function.& nbsp;

Since

\[ [x-2] = \begin{cases} -2, & amp; 0 \le x < 1,\\ -1, & amp; 1 \le x < 2,\\ 0, & amp; 2 \le x < 3,\\ 1, & amp; 3 \le x < 4,\\ 2, & amp; 4 \le x < 5, \end{cases} \]

the integral can be written as

\[ \int_{0}^{5}[x-2]\,dx = \int_{0}^{1}(-2)\,dx + \int_{1}^{2}(-1)\,dx + \int_{2}^{3}0\,dx + \int_{3}^{4}1\,dx + \int_{4}^{5}2\,dx. \]

Step 2: Evaluate each integral.

\[ =(-2)(1)+(-1)(1)+0(1)+(1)(1)+(2)(1). \] \[ =-2-1+0+1+2=0. \]

Hence,

\[ \boxed{\int_{0}^{5}[x-2]\,dx=0.} \]

Therefore, the correct option is \[ \boxed{(A)}. \]