Question

Evaluate $\displaystyle \int_{-2}^{2} [x]\,dx$, where $[x]$ denotes the greatest integer function

• A. $2$
• B. $4$
• C. $-2$
• D. $0$

Hint:

For greatest integer functions, always split the interval at integer points.

Answer 1

The Correct Option is C

Step 1: Understanding the greatest integer function.
The function $[x]$ takes constant integer values in each interval $[n,n+1)$.
Step 2: Breaking the interval.
\[ \int_{-2}^{2} [x]\,dx = \int_{-2}^{-1}(-2)\,dx + \int_{-1}^{0}(-1)\,dx + \int_{0}^{1}(0)\,dx + \int_{1}^{2}(1)\,dx \]
Step 3: Evaluating each integral.
\[ = (-2)(1) + (-1)(1) + 0(1) + 1(1) \] \[ = -2 - 1 + 0 + 1 = -2 \]
Step 4: Conclusion.
\[ \int_{-2}^{2} [x]\,dx = -2 \]