`int_(0)^(3)|2-x|dx` equals

Options

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

Answer

To solve the integral \( \int_{0}^{3} |2 - x| \, dx \), we need to analyze the expression inside the absolute value, \( |2 - x| \).

### Step 1: Determine where the expression changes sign
The expression \( 2 - x \) is positive when \( x < 2 \) and negative when \( x > 2 \). Therefore, we can break the integral into two parts based on the point where the expression changes sign (at \( x = 2 \)).

### Step 2: Split the integral
We can express the integral as:
\[ \int_{0}^{3} |2 - x| \, dx = \int_{0}^{2} (2 - x) \, dx + \int_{2}^{3} (-(2 - x)) \, dx \]
This is because:
- For \( x \) in the interval \( [0, 2] \), \( |2 - x| = 2 - x \).
- For \( x \) in the interval \( [2, 3] \), \( |2 - x| = -(2 - x) = x - 2 \).

### Step 3: Evaluate the first integral
Now, we evaluate the first integral:
\[ \int_{0}^{2} (2 - x) \, dx \]
Calculating this:
\[ = \left[ 2x - \frac{x^2}{2} \right]_{0}^{2} = \left( 2(2) - \frac{(2)^2}{2} \right) - \left( 2(0) - \frac{(0)^2}{2} \right) \]
\[ = \left( 4 - 2 \right) - 0 = 2 \]

### Step 4: Evaluate the second integral
Next, we evaluate the second integral:
\[ \int_{2}^{3} (x - 2) \, dx \]
Calculating this:
\[ = \left[ \frac{x^2}{2} - 2x \right]_{2}^{3} = \left( \frac{(3)^2}{2} - 2(3) \right) - \left( \frac{(2)^2}{2} - 2(2) \right) \]
\[ = \left( \frac{9}{2} - 6 \right) - \left( 2 - 4 \right) \]
\[ = \left( \frac{9}{2} - \frac{12}{2} \right) - (-2) = \left( -\frac{3}{2} \right) + 2 = -\frac{3}{2} + \frac{4}{2} = \frac{1}{2} \]

### Step 5: Combine the results
Now, we combine the results of both integrals:
\[ \int_{0}^{3} |2 - x| \, dx = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \]

### Final Answer
Thus, the value of the integral \( \int_{0}^{3} |2 - x| \, dx \) is:
\[ \frac{5}{2} \]