Find the area bounded by the curve `y=e^(-x)` the X-axis and the Y-axis.

Explanation

To find the area bounded by the curve \( y = e^{-x} \), the X-axis, and the Y-axis, we can follow these steps:

### Step 1: Understand the boundaries
The area we are looking for is bounded by:
- The curve \( y = e^{-x} \)
- The X-axis (where \( y = 0 \))
- The Y-axis (where \( x = 0 \))

The area is located in the first quadrant, starting from \( x = 0 \) and extending to \( x = \infty \).

### Step 2: Set up the integral
The area under the curve from \( x = 0 \) to \( x = \infty \) can be calculated using the definite integral:
\[ \text{Area} = \int_{0}^{\infty} e^{-x} \, dx \]

### Step 3: Evaluate the integral
To evaluate the integral, we first find the antiderivative of \( e^{-x} \):
\[ \int e^{-x} \, dx = -e^{-x} + C \]
Now we can apply the limits from \( 0 \) to \( \infty \):
\[ \text{Area} = \left[-e^{-x}\right]_{0}^{\infty} \]

### Step 4: Calculate the limits
Now we substitute the limits into the expression:
1. Evaluate the upper limit as \( x \) approaches infinity:
\[ \lim_{x \to \infty} -e^{-x} = -0 = 0 \]
2. Evaluate the lower limit at \( x = 0 \):
\[ -e^{0} = -1 \]

### Step 5: Find the area
Now we can calculate the area:
\[ \text{Area} = 0 - (-1) = 1 \]

Thus, the area bounded by the curve \( y = e^{-x} \), the X-axis, and the Y-axis is \( 1 \) square unit.

### Final Answer
The area bounded by the curve \( y = e^{-x} \), the X-axis, and the Y-axis is \( 1 \) square unit.

---