Question Which of the following is the best method to find the volume of the solid of revolution obtained by rotating the bounded by the graph of $$ y=7 \sqrt{x+5} $$, the $$ x $$-axis, the $$ y $$-axis, and the line $$ x=4 $$ about the line $$ y=-8 $$ ?

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To determine the best method for finding the volume of the solid of revolution obtained by rotating the region bounded by $$ y = 7\sqrt{x+5} $$, the $$ x $$-axis, the $$ y $$-axis, and the line $$ x = 4 $$ about the line $$ y = -8 $$, let's analyze the situation step-by-step.

### 1. Understanding the Bounded Region
- **Graph of $$ y = 7\sqrt{x+5} $$:** This function is defined for $$ x \geq -5 $$. For the given bounds:
  - At $$ x = 0 $$, $$ y = 7\sqrt{5} $$.
  - At $$ x = 4 $$, $$ y = 7\sqrt{9} = 21 $$.
- **Bounding Lines:**
  - $$ y $$-axis: $$ x = 0 $$.
  - $$ x = 4 $$.
  - $$ x $$-axis: $$ y = 0 $$.

So, the region of interest lies between $$ x = 0 $$ and $$ x = 4 $$, and between $$ y = 0 $$ and $$ y = 7\sqrt{x+5} $$.

### 2. Choosing the Method of Integration
Since we're rotating the region about the horizontal line $$ y = -8 $$, the **washer method** (a variation of the disk method) is most appropriate. This method involves slicing the region perpendicular to the axis of rotation (horizontal slices) and calculating the volume of each washer-shaped slice.

### 3. Setting Up the Washer Method
- **Outer Radius ($$ R(x) $$):** Distance from the axis of rotation $$ y = -8 $$ to the curve $$ y = 7\sqrt{x+5} $$:
  $$
  R(x) = 7\sqrt{x+5} + 8
  $$
- **Inner Radius ($$ r(x) $$):** Distance from the axis of rotation $$ y = -8 $$ to the $$ x $$-axis ($$ y = 0 $$):
  $$
  r(x) = 8
  $$
  
- **Volume Integral:**
  The volume $$ V $$ is given by:
  $$
  V = \pi \int_{0}^{4} \left[ (R(x))^2 - (r(x))^2 \right] dx = \pi \int_{0}^{4} \left[ (7\sqrt{x+5} + 8)^2 - 8^2 \right] dx
  $$

### Conclusion
The **washer method** is the most straightforward and efficient approach for this problem. It accounts for the distances from the curve and the boundary to the axis of rotation and facilitates setting up the integral for the volume.

**Final Answer:**

Use the washer method by integrating π[(7√(x + 5) + 8)² – 8²] with respect to x from x = 0 to x = 4. In other words,

Volume = π ∫₀⁴ [(7√(x+5)+8)^2 − 8^2] dx
Use the washer method by integrating π[(7√(x + 5) + 8)² – 8²] with respect to x from x = 0 to x = 4.

Question Which of the following is the best method to find the volume of the solid of revolution obtained by rotating the bounded by the graph of \( y=7 \sqrt{x+5} \), the \( x \)-axis, the \( y \)-axis, and the line \( x=4 \) about the line \( y=-8 \) ?

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