Question
Find the volume of a solid of revolution formed by revolving the region bounded above by the graph of
below by the -axis over the interval about the line . Enter an exact value in terms of .

Ask by Rogers Watson. in the United States

Jan 19,2025

Question

Question
Find the volume of a solid of revolution formed by revolving the region bounded above by the graph of
below by the -axis over the interval about the line . Enter an exact value in terms of .

Ask by Rogers Watson. in the United States

Jan 19,2025

UpStudy AI · Accepted Answer

To find the volume of the solid of revolution formed by revolving the region bounded above by $$ f(x) = x + 4 $$ and below by the $$ x $$-axis over the interval $$[1, 4]$$ about the line $$ y = -2 $$, we'll use the **washer method**.

### Step-by-Step Solution:

1. **Identify the Radii:**
   - **Outer Radius ($$ R $$)**: The distance from the curve $$ f(x) = x + 4 $$ to the line $$ y = -2 $$:
     $$
     R = (x + 4) - (-2) = x + 6
     $$
   - **Inner Radius ($$ r $$)**: The distance from the $$ x $$-axis ($$ y = 0 $$) to the line $$ y = -2 $$:
     $$
     r = 0 - (-2) = 2
     $$

2. **Volume of a Washer:**
   The volume $$ dV $$ of a washer with thickness $$ dx $$ is:
   $$
   dV = \pi \left(R^2 - r^2\right) dx = \pi \left((x + 6)^2 - 2^2\right) dx = \pi \left((x + 6)^2 - 4\right) dx
   $$

3. **Integrate Over the Interval $$[1, 4]$$:**
   $$
   V = \pi \int_{1}^{4} \left((x + 6)^2 - 4\right) dx
   $$
   
   Expand and simplify the integrand:
   $$
   (x + 6)^2 - 4 = x^2 + 12x + 36 - 4 = x^2 + 12x + 32
   $$
   
   Now integrate:
   $$
   V = \pi \left[ \frac{x^3}{3} + 6x^2 + 32x \right]_{1}^{4}
   $$
   
   Evaluate the definite integral:
   $$
   V = \pi \left( \left(\frac{4^3}{3} + 6 \cdot 4^2 + 32 \cdot 4\right) - \left(\frac{1^3}{3} + 6 \cdot 1^2 + 32 \cdot 1\right) \right)
   $$
   
   Simplify the calculations:
   $$
   V = \pi \left( \left(\frac{64}{3} + 96 + 128\right) - \left(\frac{1}{3} + 6 + 32\right) \right) = \pi \left(\frac{736}{3} - \frac{115}{3}\right) = \pi \cdot \frac{621}{3} = 207\pi
   $$

### Final Answer:
The volume is $$ 207\pi $$.
The volume of the solid is $$ 207\pi $$.

Question
Find the volume of a solid of revolution formed by revolving the region bounded above by the graph of
below by the -axis over the interval about the line . Enter an exact value in terms of .

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Question Find the volume of a solid of revolution formed by revolving the region bounded above by the graph of below by the -axis over the interval about the line . Enter an exact value in terms of .

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Question
Find the volume of a solid of revolution formed by revolving the region bounded above by the graph of
below by the -axis over the interval about the line . Enter an exact value in terms of .