Find the volume of the solid obtained by rotating the region enclosed by the graphs y = e^x , y = 0, x = 0, and x =7 about y = -6

To find the volume of the solid obtained by rotating the given region about the line \( y = -6 \), we use the method of cylindrical shells. The region is bounded by the curves \( y = e^x \), \( y = 0 \), \( x = 0 \), and \( x = 7 \).

### Step 1: Set up the cylindrical shells formula
The formula for the volume using cylindrical shells is:

\[
V = 2\pi \int_{a}^{b} (radius) \times (height) \, dx
\]

Here:
- **Radius:** The distance from the curve \( y = e^x \) to the axis of rotation \( y = -6 \).
- **Height:** The height of each cylindrical shell, which is the function \( y = e^x \).

### Step 2: Determine the radius
The radius is the vertical distance between \( y = e^x \) and \( y = -6 \):

\[
\text{Radius} = (e^x - (-6)) = e^x + 6
\]

### Step 3: Determine the height
The height of each shell is simply the value of the function:

\[
\text{Height} = e^x
\]

### Step 4: Set up the integral
The limits of integration are from \( x = 0 \) to \( x = 7 \), so the volume \( V \) is:

\[
V = 2\pi \int_{0}^{7} (e^x + 6) \cdot e^x \, dx
\]

### Step 5: Simplify the integrand
Simplify the integrand by multiplying out the terms:

\[
V = 2\pi \int_{0}^{7} (e^{2x} + 6e^x) \, dx
\]

### Step 6: Integrate
Now, integrate each term separately:

\[
V = 2\pi \left[\int_{0}^{7} e^{2x} \, dx + 6\int_{0}^{7} e^x \, dx \right]
\]

The first integral is:

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

The second integral is:

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

So the volume integral becomes:

\[
V = 2\pi \left[\frac{1}{2}e^{2x} \Big|_0^7 + 6e^x \Big|_0^7 \right]
\]

### Step 7: Evaluate the definite integrals
Evaluate at the bounds \( x = 7 \) and \( x = 0 \):

\[
\frac{1}{2}e^{2(7)} - \frac{1}{2}e^{2(0)} = \frac{1}{2}e^{14} - \frac{1}{2}
\]
\[
6(e^{7} - e^{0}) = 6(e^{7} - 1)
\]

Thus, the volume is:

\[
V = 2\pi \left[\frac{1}{2}(e^{14} - 1) + 6(e^7 - 1)\right]
\]

Simplifying further:

\[
V = \pi \left(e^{14} - 1 + 12e^7 - 12\right)
\]

\[
V = \pi \left(e^{14} + 12e^7 - 13\right)
\]

So, the volume of the solid is:

\[
\boxed{V = \pi \left(e^{14} + 12e^7 - 13\right)}
\]

Would you like further details or have any questions?

### Relative Questions:
1. How would the integral change if the region were rotated about the line \( y = 6 \)?
2. What is the significance of choosing the cylindrical shells method over the disk/washer method for this problem?
3. How would you compute the volume if the region were rotated about the y-axis instead?
4. What would happen to the volume if the upper bound for \( x \) were changed from 7 to 8?
5. Can you generalize the formula for rotating about any horizontal line \( y = c \)?

### Tip:
Always consider the axis of rotation carefully when setting up the radius for cylindrical shells or washers; it will affect the entire setup and final answer.

Learn how to find the volume of the solid obtained by rotating the region enclosed by the graphs y = e^x, y = 0, x = 0, and x = 7 about the line y = -6 using cylindrical shells method in calculus. Detailed step-by-step solution included.

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