What is \(\mathop \smallint \limits_{ - 2}^2 {\rm{x\;dx}} - \mathop \smallint \limits_{ - 2}^2 \left[ {\rm{x}} \right]{\rm{dx}}\) equal to, where [⋅] is the greatest integer function?
2
Understanding the Problem: Definite Integrals
The question asks us to evaluate the difference between two definite integrals over the interval [-2, 2]. The first integral is of the function \(\rm{f(x) = x}\), and the second integral is of the greatest integer function, denoted by \(\left[ {\rm{x}} \right]\). We need to calculate each integral separately and then find their difference.
Evaluating the First Integral: \(\mathop \smallint \limits_{ - 2}^2 {\rm{x\;dx}}\)
The first integral is \(\mathop \smallint \limits_{ - 2}^2 {\rm{x\;dx}}\). The function \(\rm{f(x) = x}\) is an odd function because \(\rm{f(-x) = -x = -f(x)}\). We are integrating over a symmetric interval [-2, 2].
A property of definite integrals states that if a function \(\rm{f(x)}\) is odd, then \(\mathop \smallint \limits_{ - a}^a {\rm{f(x)\;dx}} = 0\).
Alternatively, we can calculate it directly:
The antiderivative of \(\rm{x}\) is \(\frac{{{{\rm{x}}^2}}}{2}\).
So, \(\mathop \smallint \limits_{ - 2}^2 {\rm{x\;dx}} = \left[ {\frac{{{{\rm{x}}^2}}}{2}} \right]_{ - 2}^2 = \frac{{{{\left( 2 \right)}^2}}}{2} - \frac{{{{\left( { - 2} \right)}^2}}}{2} = \frac{4}{2} - \frac{4}{2} = 2 - 2 = 0\).
Thus, the value of the first integral is 0.
Evaluating the Second Integral: \(\mathop \smallint \limits_{ - 2}^2 \left[ {\rm{x}} \right]{\rm{dx}}\)
The second integral involves the greatest integer function, \(\left[ {\rm{x}} \right]\). The greatest integer function gives the largest integer less than or equal to \(\rm{x}\). It is a step function that is constant over intervals of the form \([\rm{n}, \rm{n}+1)\), where \(\rm{n}\) is an integer.
To evaluate the definite integral of the greatest integer function over the interval [-2, 2], we need to split the interval into sub-intervals where \(\left[ {\rm{x}} \right]\) is constant:
- For \(-2 \le \rm{x} < -1\), \(\left[ {\rm{x}} \right] = -2\)
- For \(-1 \le \rm{x} < 0\), \(\left[ {\rm{x}} \right] = -1\)
- For \(0 \le \rm{x} < 1\), \(\left[ {\rm{x}} \right] = 0\)
- For \(1 \le \rm{x} < 2\), \(\left[ {\rm{x}} \right] = 1\)
- At \(\rm{x} = 2\), \(\left[ {\rm{x}} \right] = 2\). The value at a single point does not affect the definite integral.
So, we can write the integral as the sum of integrals over these intervals:
\(\mathop \smallint \limits_{ - 2}^2 \left[ {\rm{x}} \right]{\rm{dx}} = \mathop \smallint \limits_{ - 2}^{ - 1} \left[ {\rm{x}} \right]{\rm{dx}} + \mathop \smallint \limits_{ - 1}^0 \left[ {\rm{x}} \right]{\rm{dx}} + \mathop \smallint \limits_0^1 \left[ {\rm{x}} \right]{\rm{dx}} + \mathop \smallint \limits_1^2 \left[ {\rm{x}} \right]{\rm{dx}}\)
Substitute the constant values of \(\left[ {\rm{x}} \right]\) in each interval:
\(= \mathop \smallint \limits_{ - 2}^{ - 1} (-2){\rm{dx}} + \mathop \smallint \limits_{ - 1}^0 (-1){\rm{dx}} + \mathop \smallint \limits_0^1 (0){\rm{dx}} + \mathop \smallint \limits_1^2 (1){\rm{dx}}\)
Now, evaluate each integral:
- \(\mathop \smallint \limits_{ - 2}^{ - 1} (-2){\rm{dx}} = [-2{\rm{x}}]_{ - 2}^{ - 1} = (-2)(-1) - (-2)(-2) = 2 - 4 = -2\)
- \(\mathop \smallint \limits_{ - 1}^0 (-1){\rm{dx}} = [-{\rm{x}}]_{ - 1}^0 = (-0) - (-(-1)) = 0 - 1 = -1\)
- \(\mathop \smallint \limits_0^1 (0){\rm{dx}} = [0]_{ 0}^1 = 0 - 0 = 0\)
- \(\mathop \smallint \limits_1^2 (1){\rm{dx}} = [{\rm{x}}]_{ 1}^2 = (2) - (1) = 1\)
Summing these results to get the value of the second integral:
\(\mathop \smallint \limits_{ - 2}^2 \left[ {\rm{x}} \right]{\rm{dx}} = -2 + (-1) + 0 + 1 = -3 + 1 = -2\)
Thus, the value of the second integral is -2.
Calculating the Difference
The problem asks for the value of \(\mathop \smallint \limits_{ - 2}^2 {\rm{x\;dx}} - \mathop \smallint \limits_{ - 2}^2 \left[ {\rm{x}} \right]{\rm{dx}}\).
We found that \(\mathop \smallint \limits_{ - 2}^2 {\rm{x\;dx}} = 0\) and \(\mathop \smallint \limits_{ - 2}^2 \left[ {\rm{x}} \right]{\rm{dx}} = -2\).
So, the difference is \(0 - (-2) = 0 + 2 = 2\).
The value of the expression is 2.
Let's summarize the steps:
| Integral | Function Type / Method | Calculation | Result |
|---|---|---|---|
| \(\mathop \smallint \limits_{ - 2}^2 {\rm{x\;dx}}\) | Odd function over symmetric interval | \(\left[ {\frac{{{{\rm{x}}^2}}}{2}} \right]_{ - 2}^2 = \frac{4}{2} - \frac{4}{2}\) | 0 |
| \(\mathop \smallint \limits_{ - 2}^2 \left[ {\rm{x}} \right]{\rm{dx}}\) | Greatest Integer Function, split intervals | \(\mathop \smallint \limits_{ - 2}^{ - 1} (-2){\rm{dx}} + \mathop \smallint \limits_{ - 1}^0 (-1){\rm{dx}} + \mathop \smallint \limits_0^1 (0){\rm{dx}} + \mathop \smallint \limits_1^2 (1){\rm{dx}}\) \(= (-2) + (-1) + 0 + 1\) |
-2 |
Difference = Result of first integral - Result of second integral = \(0 - (-2) = 2\).
The final answer is 2.
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