Find the area bounded by `y = xe^|x|` and lines `|x|=1,y=0`.

Options

A. 4
B. 6
C. 1
D. 2

Explanation

To find the area bounded by the curve \( y = x e^{|x|} \) and the lines \( |x| = 1 \) and \( y = 0 \), we can follow these steps:

### Step 1: Understand the boundaries
The lines \( |x| = 1 \) imply that we are looking at the region between \( x = -1 \) and \( x = 1 \). The line \( y = 0 \) is the x-axis.

### Step 2: Analyze the function
The function \( y = x e^{|x|} \) can be split into two cases based on the definition of the absolute value:
- For \( x \geq 0 \): \( y = x e^{x} \)
- For \( x < 0 \): \( y = x e^{-x} \)

### Step 3: Sketch the graph
Sketch the graph of \( y = x e^{|x|} \) from \( x = -1 \) to \( x = 1 \). The graph is symmetric about the y-axis because \( e^{|x|} \) is an even function.

### Step 4: Set up the integral for area
Since the area is symmetric about the y-axis, we can calculate the area from \( 0 \) to \( 1 \) and then double it:
\[ \text{Area} = 2 \int_{0}^{1} x e^{x} \, dx \]

### Step 5: Evaluate the integral
To evaluate the integral \( \int x e^{x} \, dx \), we can use integration by parts. Let:
- \( u = x \) → \( du = dx \)
- \( dv = e^{x} dx \) → \( v = e^{x} \)

Using integration by parts:
\[ \int u \, dv = uv - \int v \, du \]
\[ \int x e^{x} \, dx = x e^{x} - \int e^{x} \, dx = x e^{x} - e^{x} + C \]

### Step 6: Apply the limits
Now, we can evaluate the definite integral:
\[ \int_{0}^{1} x e^{x} \, dx = \left[ x e^{x} - e^{x} \right]_{0}^{1} \]
Calculating this:
\[ = \left[ 1 \cdot e^{1} - e^{1} \right] - \left[ 0 \cdot e^{0} - e^{0} \right] \]
\[ = (e - e) - (0 - 1) = 0 + 1 = 1 \]

### Step 7: Calculate the total area
Now, substituting back into the area formula:
\[ \text{Area} = 2 \cdot 1 = 2 \]

### Final Answer
The area bounded by the curve \( y = x e^{|x|} \) and the lines \( |x| = 1 \) and \( y = 0 \) is \( 2 \) square units.

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