The area of the region bounded by the curves `y = xe^x, y = e^x` and the lines `x = +-1,` is equal to

Options

A. (a)'e-(1)/(e)' sq. units
B. (b)`e + (2)/(e)` sq. unit
C. (c)`e - (3)/(e)` sq. unit
D. (d)`e + (3)/(e)` sq. unit

Answer

To find the area of the region bounded by the curves \( y = xe^x \), \( y = e^x \), and the lines \( x = -1 \) and \( x = 1 \), we will follow these steps:

### Step 1: Find the points of intersection
To find the points of intersection between the curves \( y = xe^x \) and \( y = e^x \), we set them equal to each other:

\[ xe^x = e^x \]

Dividing both sides by \( e^x \) (since \( e^x \neq 0 \)) gives:

\[ x = 1 \]

Now, we also need to check the value at \( x = -1 \):

\[ y = e^{-1} = \frac{1}{e} \]

### Step 2: Determine the area between the curves
The area \( A \) between the curves from \( x = -1 \) to \( x = 1 \) can be expressed as:

\[ A = \int_{-1}^{1} (e^x - xe^x) \, dx \]

### Step 3: Simplify the integral
We can rewrite the integral:

\[ A = \int_{-1}^{1} e^x \, dx - \int_{-1}^{1} xe^x \, dx \]

### Step 4: Calculate the first integral
The first integral can be calculated as follows:

\[ \int e^x \, dx = e^x \]

Thus,

\[ \int_{-1}^{1} e^x \, dx = e^1 - e^{-1} = e - \frac{1}{e} \]

### Step 5: Calculate the second integral
For the second integral, we use integration by parts. Let:

- \( u = x \) → \( du = dx \)
- \( dv = e^x dx \) → \( v = e^x \)

Using integration by parts:

\[ \int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x \]

Now, we evaluate this from \( -1 \) to \( 1 \):

\[ \left[ x e^x - e^x \right]_{-1}^{1} = \left[ 1 \cdot e^1 - e^1 \right] - \left[ -1 \cdot e^{-1} - e^{-1} \right] \]

Calculating this gives:

\[ (e - e) - \left( -\frac{1}{e} - \frac{1}{e} \right) = 0 + \frac{2}{e} = \frac{2}{e} \]

### Step 6: Combine the results
Now we can combine the results of the two integrals:

\[ A = \left( e - \frac{1}{e} \right) - \frac{2}{e} \]

This simplifies to:

\[ A = e - \frac{1}{e} - \frac{2}{e} = e - \frac{3}{e} \]

### Final Result
Thus, the area of the region bounded by the curves is:

\[ A = e - \frac{3}{e} \]