Let 
\[R\] be the region enclosed by the line 
\[x=-4\], the line 
\[y=1\], the line 
\[y=4\], and the curve 
\[y=e^x\].
Line x = negative 4 and curve y = e to the x power are graphed. The curve moves along the negative x axis, then moves upward with increasing steepness. It intersects the line at (negative 4, 0), then moves upward to the right of it. The area enclosed by the two plots between y = 1 and y = 4 is shaded, representing region R.
\[y\]
\[x\]
\[\blueD{y=e^x}\]
\[\small -4\]
\[\large R\]
\[\small 1\]
\[\small 4\]
A solid is generated by rotating 
\[R\] about the line 
\[x=-4\].
Which one of the definite integrals gives the volume of the solid?
Choose 1 answer:
Choose 1 answer:

To find the volume of the solid generated by rotating the region \( R \) around the line \( x = -4 \), we need to set up the integral using the method of cylindrical shells. 

### Step 1: Determine the formula for the volume using cylindrical shells
The formula for the volume using cylindrical shells when rotating around a vertical line (like \( x = -4 \)) is given by:

\[
V = 2\pi \int_{y_{\text{min}}}^{y_{\text{max}}} \text{(radius)} \times \text{(height)} \, dy
\]

### Step 2: Identify the radius and height
- **Radius**: The distance from the axis of rotation \( x = -4 \) to a point on the curve is given by the horizontal distance between \( x = -4 \) and \( x \) on the curve. Thus, the radius \( r(y) = |x + 4| = | \ln y + 4 | = \ln y + 4 \).

- **Height**: The height of the shell is the difference between the \( y \)-values, which is \( y \) (since we are using \( y \) as the independent variable in this approach).

### Step 3: Determine the limits of integration
The region \( R \) is enclosed between \( y = 1 \) and \( y = 4 \), so the limits of integration are from \( y = 1 \) to \( y = 4 \).

### Step 4: Set up the integral
Thus, the volume of the solid is:

\[
V = 2\pi \int_{1}^{4} (\ln y + 4) \cdot y \, dy
\]

However, the problem asks which of the definite integrals corresponds to the correct setup. The integral expression that matches the setup described above would be the following:

\[
V = 2\pi \int_{1}^{4} (\ln y + 4) \cdot y \, dy
\]

This represents the correct definite integral to calculate the volume of the solid generated by rotating the region \( R \) about the line \( x = -4 \).

Would you like more details on this solution, or do you have any questions?

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**Relative Questions**:
1. How would the integral change if the rotation were around \( y = 0 \) instead?
2. What is the significance of the height function in the method of cylindrical shells?
3. How would you set up the integral if the region \( R \) was rotated around the y-axis instead of \( x = -4 \)?
4. Can the disk/washer method be applied to this problem? If yes, how?
5. What if the curve \( y = e^x \) was replaced with \( y = x^2 \); how would the integral change?

**Tip**: Always visualize the region and the axis of rotation to help determine the correct method and setup for the integral.

Learn how to calculate the volume of a solid generated by rotating the region enclosed by x = -4, y = 1, y = 4, and y = e^x around the line x = -4. Detailed solution using the method of cylindrical shells in integral calculus.

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