Question:
If $[x]$ is the greatest integer less than or equal to x, then $\int_{-3}^{3}[x]dx=$ ________.
If $[x]$ is the greatest integer less than or equal to x, then $\int_{-3}^{3}[x]dx=$ ________.
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Integrate $[x]$ by finding the area of the rectangles formed by the steps.
Updated On: Jun 26, 2026
- -3
- -6
- -4
- -2
- 0
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The Correct Option is A
Solution and Explanation
Step 1: Concept
The integral of a step function is the sum of (value $\times$ width) for each interval.
Step 2: Meaning
Intervals: $[-3,-2], [-2,-1], [-1,0], [0,1], [1,2], [2,3]$ with widths of 1 unit.
Step 3: Analysis
Values: $(-3 \times 1) + (-2 \times 1) + (-1 \times 1) + (0 \times 1) + (1 \times 1) + (2 \times 1)$.
Step 4: Conclusion
$-3 - 2 - 1 + 0 + 1 + 2 = -3$. Final Answer: (A)
The integral of a step function is the sum of (value $\times$ width) for each interval.
Step 2: Meaning
Intervals: $[-3,-2], [-2,-1], [-1,0], [0,1], [1,2], [2,3]$ with widths of 1 unit.
Step 3: Analysis
Values: $(-3 \times 1) + (-2 \times 1) + (-1 \times 1) + (0 \times 1) + (1 \times 1) + (2 \times 1)$.
Step 4: Conclusion
$-3 - 2 - 1 + 0 + 1 + 2 = -3$. Final Answer: (A)
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