Use the disk method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the *x*-axis.

*y* = *e*−*x*,    *y* = 0,    *x* = 0,    *x* = 3

To find the volume of the solid generated by revolving the region bounded by the curves about the \( x \)-axis, we can use the disk method.

### Given:
- \( y = e^{-x} \)
- \( y = 0 \) (the x-axis)
- \( x = 0 \)
- \( x = 3 \)

### Disk Method Formula:
The volume \( V \) of the solid generated by revolving the region about the \( x \)-axis is given by:

\[
V = \pi \int_{a}^{b} \left[ f(x) \right]^2 \, dx
\]

where \( f(x) \) represents the function defining the curve, and \( a \) and \( b \) are the limits of integration.

### Step-by-Step Solution:
1. **Identify the function**: The curve is \( y = e^{-x} \).
   
2. **Set up the integral**: The bounds are from \( x = 0 \) to \( x = 3 \). The volume is given by:
   
\[
V = \pi \int_{0}^{3} \left[ e^{-x} \right]^2 \, dx
\]

3. **Simplify the integrand**:
   
\[
V = \pi \int_{0}^{3} e^{-2x} \, dx
\]

4. **Integrate** \( e^{-2x} \):
   
The antiderivative of \( e^{-2x} \) is:

\[
\int e^{-2x} \, dx = -\frac{1}{2} e^{-2x}
\]

So,

\[
V = \pi \left[ -\frac{1}{2} e^{-2x} \right]_{0}^{3}
\]

5. **Evaluate the definite integral**:

\[
V = \pi \left[ -\frac{1}{2} e^{-2(3)} - \left( -\frac{1}{2} e^{-2(0)} \right) \right]
\]

\[
V = \pi \left[ -\frac{1}{2} e^{-6} + \frac{1}{2} e^{0} \right]
\]

\[
V = \pi \left( \frac{1}{2} - \frac{1}{2} e^{-6} \right)
\]

\[
V = \frac{\pi}{2} \left( 1 - e^{-6} \right)
\]

### Final Answer:
The volume of the solid is:

\[
V = \frac{\pi}{2} \left( 1 - e^{-6} \right) \text{ cubic units}
\]

---

Do you want any further details or explanations?  

Here are some related questions you might find interesting:

1. What happens to the volume if we rotate around the y-axis instead?
2. How would the volume change if the upper bound of x was 2 instead of 3?
3. Can the washer method be used to find the volume in this problem?
4. What if the region was rotated about the line \( y = 1 \) instead of the x-axis?
5. How do you find the centroid of the region before rotation?
6. What is the significance of the exponential decay in the function \( y = e^{-x} \)?
7. How does changing the base of the exponential function affect the volume?
8. How would you approach the problem if the function was \( y = e^{-2x} \) instead?

**Tip:** When setting up integrals for volumes of solids of revolution, always sketch the region and identify the axis of rotation to decide whether to use the disk or washer method.

Learn how to calculate the volume of a solid of revolution using the disk method with the functions y = e^(-x), y = 0, and bounds x = 0 to x = 3. Step-by-step solution provided.

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