The value of `int_(-3)^(3) |x| dx` is

Options

A. 3
B. 6
C. 9
D. none

Answer

To solve the integral \( \int_{-3}^{3} |x| \, dx \), we will break it into two parts based on the definition of the absolute value function.

### Step 1: Break the integral into two parts
The absolute value function \( |x| \) can be defined as:
- \( |x| = -x \) when \( x < 0 \)
- \( |x| = x \) when \( x \geq 0 \)

Thus, we can split the integral at 0:
\[ \int_{-3}^{3} |x| \, dx = \int_{-3}^{0} |x| \, dx + \int_{0}^{3} |x| \, dx \]

### Step 2: Evaluate the first integral
For the first integral, where \( x < 0 \):
\[ \int_{-3}^{0} |x| \, dx = \int_{-3}^{0} -x \, dx \]
Now, we can integrate:
\[ = -\left[ \frac{x^2}{2} \right]_{-3}^{0} \]
Calculating this:
\[ = -\left( \frac{0^2}{2} - \frac{(-3)^2}{2} \right) = -\left( 0 - \frac{9}{2} \right) = \frac{9}{2} \]

### Step 3: Evaluate the second integral
For the second integral, where \( x \geq 0 \):
\[ \int_{0}^{3} |x| \, dx = \int_{0}^{3} x \, dx \]
Now, we can integrate:
\[ = \left[ \frac{x^2}{2} \right]_{0}^{3} \]
Calculating this:
\[ = \left( \frac{3^2}{2} - \frac{0^2}{2} \right) = \frac{9}{2} \]

### Step 4: Combine the results
Now we can combine the results from both integrals:
\[ \int_{-3}^{3} |x| \, dx = \int_{-3}^{0} |x| \, dx + \int_{0}^{3} |x| \, dx = \frac{9}{2} + \frac{9}{2} = 9 \]

### Final Answer
Thus, the value of the integral \( \int_{-3}^{3} |x| \, dx \) is \( 9 \).

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