Question

If $[x]$ is the greatest integer less than or equal to x, then $\int_{-3}^{3}[x]dx=$ ________.

• A. -3
• B. -6
• C. -4
• D. -2
• E. 0

Hint:

Integrate $[x]$ by finding the area of the rectangles formed by the steps.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
We need to evaluate a definite integral of the greatest integer function, which is a step function. Because the function is piecewise constant, we must break the interval of integration into subintervals where the function \([x]\) has a constant value.
Step 2: Key Formula or Approach
The greatest integer function \([x]\) is constant between any two consecutive integers. We will use the property of definite integrals that allows splitting the interval:
\[ \int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx \] We will split the interval \([-2, 1]\) at the integers it contains, which are -1 and 0.
Step 3: Detailed Explanation
1. Split the integral at integer points.
The interval of integration is from -2 to 1. The integers within this range where the value of \([x]\) changes are -1 and 0. So we split the integral into three parts:
\[ \int_{-2}^1 [x] dx = \int_{-2}^{-1} [x] dx + \int_{-1}^{0} [x] dx + \int_{0}^{1} [x] dx \] 2. Determine the value of \([x]\) in each subinterval.
- For \(-2 \le x<-1\), the value of \([x]\) is -2.
- For \(-1 \le x<0\), the value of \([x]\) is -1.
- For \(0 \le x<1\), the value of \([x]\) is 0.
Note: The value of the integrand at a single point (like at x=-1) does not affect the value of the definite integral.
3. Evaluate each integral.
Substitute the constant values of \([x]\) into each integral:
\[ \int_{-2}^1 [x] dx = \int_{-2}^{-1} (-2) dx + \int_{-1}^{0} (-1) dx + \int_{0}^{1} (0) dx \] Now, integrate each constant:
\[ = [-2x]_{-2}^{-1} + [-x]_{-1}^{0} + [0x]_{0}^{1} \] Apply the limits:
\[ = (-2(-1) - (-2)(-2)) + (-(0) - (-(-1))) + (0) \] \[ = (2 - 4) + (0 - 1) + 0 \] \[ = -2 - 1 + 0 \] \[ = -3 \] Step 4: Final Answer
The value of the integral is -3.