`int_(-1)^(2)|x|dx=`

Options

A. `5/2`
B. `1/2`
C. `3/2`
D. `7/2`

Answer

To solve the integral \(\int_{-1}^{2} |x| \, dx\), we need to consider the definition of the absolute value function, which behaves differently depending on whether \(x\) is positive or negative.

### Step 1: Determine the intervals for the absolute value function
The absolute value function \(|x|\) can be expressed as:
- \(|x| = -x\) when \(x < 0\)
- \(|x| = x\) when \(x \geq 0\)

Given the limits of integration from \(-1\) to \(2\), we can break the integral into two parts:
1. From \(-1\) to \(0\) (where \(|x| = -x\))
2. From \(0\) to \(2\) (where \(|x| = x\))

### Step 2: Set up the integral
We can write the integral as:
\[ \int_{-1}^{2} |x| \, dx = \int_{-1}^{0} -x \, dx + \int_{0}^{2} x \, dx \]

### Step 3: Calculate the first integral
Now, we calculate the first integral:
\[ \int_{-1}^{0} -x \, dx \]
The antiderivative of \(-x\) is \(-\frac{x^2}{2}\). We evaluate this from \(-1\) to \(0\):
\[ \left[-\frac{x^2}{2}\right]_{-1}^{0} = \left[-\frac{0^2}{2}\right] - \left[-\frac{(-1)^2}{2}\right] = 0 - \left[-\frac{1}{2}\right] = \frac{1}{2} \]

### Step 4: Calculate the second integral
Next, we calculate the second integral:
\[ \int_{0}^{2} x \, dx \]
The antiderivative of \(x\) is \(\frac{x^2}{2}\). We evaluate this from \(0\) to \(2\):
\[ \left[\frac{x^2}{2}\right]_{0}^{2} = \left[\frac{2^2}{2}\right] - \left[\frac{0^2}{2}\right] = \frac{4}{2} - 0 = 2 \]

### Step 5: Combine the results
Now, we combine both results:
\[ \int_{-1}^{2} |x| \, dx = \frac{1}{2} + 2 = \frac{1}{2} + \frac{4}{2} = \frac{5}{2} \]

### Final Answer
Thus, the value of the integral \(\int_{-1}^{2} |x| \, dx\) is:
\[ \frac{5}{2} \]