What is the area under the curve `y=|x|+|x-1|` between x=0 and x=1?

Options

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

Answer

To find the area under the curve \( y = |x| + |x - 1| \) between \( x = 0 \) and \( x = 1 \), we will follow these steps:

### Step 1: Analyze the function
The function \( y = |x| + |x - 1| \) consists of two absolute value terms. We need to determine how these terms behave in the interval \( [0, 1] \).

### Step 2: Break down the absolute values
1. For \( x \) in the interval \( [0, 1] \):
- \( |x| = x \) (since \( x \) is non-negative)
- \( |x - 1| = 1 - x \) (since \( x - 1 \) is non-positive)

Thus, in the interval \( [0, 1] \):
\[ y = x + (1 - x) = 1 \]

### Step 3: Set up the integral
Since \( y = 1 \) is a constant function in the interval \( [0, 1] \), the area under the curve can be calculated by integrating the function from \( x = 0 \) to \( x = 1 \):
\[ \text{Area} = \int_{0}^{1} 1 \, dx \]

### Step 4: Calculate the integral
Now, we compute the integral:
\[ \int_{0}^{1} 1 \, dx = [x]_{0}^{1} = 1 - 0 = 1 \]

### Step 5: Conclusion
The area under the curve \( y = |x| + |x - 1| \) between \( x = 0 \) and \( x = 1 \) is \( 1 \) square unit.

### Final Answer
The area under the curve is \( 1 \) square unit.

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