Question

The value of \(\int_{1}^{2} [x-1]\,dx\), where \([x]\) denotes the greatest integer function in \(x\), is equal to

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

Hint:

For greatest integer functions, always split the interval at integer points where the function value changes.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The problem is to evaluate the definite integral of the greatest integer function (or floor function), \( [x-1] \). The greatest integer function, \( [y] \), gives the greatest integer less than or equal to \( y \). This function is a step function, meaning it is constant over intervals and jumps at integer values. To integrate it, we must break the integral into sub-intervals where the function's value is constant.
Step 2: Key Formula or Approach:
1. Analyze the value of the integrand \( [x-1] \) over the interval of integration \( [0, 2] \).
2. The value of \( [x-1] \) changes when \( x-1 \) is an integer.
3. Split the integral \( \int_0^2 \) into parts based on where the value of \( [x-1] \) changes.
4. Evaluate the simple integrals over these sub-intervals.
Step 3: Detailed Explanation:
Let \( f(x) = [x-1] \). We need to evaluate \( \int_0^2 f(x) dx \).
The argument of the floor function is \( x-1 \). As \( x \) goes from 0 to 2, \( x-1 \) goes from -1 to 1. The points where the value of \( [x-1] \) changes are when \( x-1 \) crosses an integer. In this range, this happens at \( x-1=0 \), which is \( x=1 \).
So, we split the interval of integration \( [0, 2] \) at \( x=1 \).
Interval 1: \( 0 \le x<1 \)
In this interval, we have \( -1 \le x-1<0 \).
The greatest integer less than or equal to a number in \( [-1, 0) \) is -1.
So, for \( x \in [0, 1) \), \( [x-1] = -1 \).
Interval 2: \( 1 \le x \le 2 \)
In this interval, we have \( 0 \le x-1 \le 1 \).
For \( 1 \le x<2 \), we have \( 0 \le x-1<1 \), so \( [x-1] = 0 \).
At the single point \( x=2 \), \( [2-1] = [1] = 1 \). The value at a single point does not affect the value of a definite integral.
Now, we split the integral:
\[ \int_0^2 [x-1] dx = \int_0^1 [x-1] dx + \int_1^2 [x-1] dx \] Substitute the constant values we found for each interval:
\[ = \int_0^1 (-1) dx + \int_1^2 (0) dx \] Evaluate the integrals:
\[ = [-x]_0^1 + [0]_1^2 \] \[ = (-(1) - (-0)) + 0 \] \[ = -1 \] The provided answer key states the correct answer is option (D), which is 1. Based on a direct evaluation of the integral as written, the result is -1. There might be a typo in the question or the answer key. For example, if the integral was \( \int_0^2 |x-1| dx \), the result would be 1. However, based on the question as written, the answer is -1. We will follow the mathematical derivation. The provided key in the source document is likely incorrect. The correct option is (B).
Step 4: Final Answer:
The value of the integral is -1. This corresponds to option (B).