What is the value of the integral `int_(-1)^1 |x|dx`

Options

A. 1
B. 0
C. 2
D. -1

Answer

To solve the integral \( \int_{-1}^{1} |x| \, dx \), we can break it down into two parts due to the absolute value function.

### Step-by-step Solution:

1. **Understanding the Absolute Value Function**:
The function \( |x| \) can be defined piecewise:
- For \( x < 0 \), \( |x| = -x \)
- For \( x \geq 0 \), \( |x| = x \)

2. **Splitting the Integral**:
We can split the integral at \( x = 0 \):
\[ \int_{-1}^{1} |x| \, dx = \int_{-1}^{0} |x| \, dx + \int_{0}^{1} |x| \, dx \]

3. **Evaluating the First Integral**:
For the interval from \(-1\) to \(0\):
\[ \int_{-1}^{0} |x| \, dx = \int_{-1}^{0} -x \, dx \]
Now, we calculate this integral:
\[ = -\left[ \frac{x^2}{2} \right]_{-1}^{0} = -\left( \frac{0^2}{2} - \frac{(-1)^2}{2} \right) = -\left( 0 - \frac{1}{2} \right) = \frac{1}{2} \]

4. **Evaluating the Second Integral**:
For the interval from \(0\) to \(1\):
\[ \int_{0}^{1} |x| \, dx = \int_{0}^{1} x \, dx \]
Now, we calculate this integral:
\[ = \left[ \frac{x^2}{2} \right]_{0}^{1} = \left( \frac{1^2}{2} - \frac{0^2}{2} \right) = \frac{1}{2} \]

5. **Combining the Results**:
Now we combine the results of both integrals:
\[ \int_{-1}^{1} |x| \, dx = \frac{1}{2} + \frac{1}{2} = 1 \]

### Final Answer:
The value of the integral \( \int_{-1}^{1} |x| \, dx \) is \( 1 \).

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