https://file.aiquickdraw.com/mathbot/file/1727228933423o9imovdk.jpg

This problem asks to find the volume of the solid obtained by rotating the region bounded by the curves \( y = x^8 \) and \( y = 1 \) about the line \( y = 4 \).

### Step-by-Step Solution:

1. **Visualizing the problem:**
   - The region is bounded by the curves \( y = x^8 \) and \( y = 1 \) over a certain range of \( x \), from \( x = 0 \) to \( x = 1 \) (since \( y = x^8 \) and \( y = 1 \) intersect at \( x = 1 \)).
   - This region is rotated about the line \( y = 4 \).

2. **Setup for Volume Using the Washer Method:**
   - The washer method involves calculating the volume of a solid by subtracting the volume of the inner solid from the outer solid.
   - The formula for volume using the washer method is:
     \[
     V = \pi \int_{a}^{b} \left[ R_{\text{outer}}(x)^2 - R_{\text{inner}}(x)^2 \right] \, dx
     \]
   where \( R_{\text{outer}}(x) \) is the distance from the rotation axis to the outer curve, and \( R_{\text{inner}}(x) \) is the distance to the inner curve.

3. **Define the Radii:**
   - The distance from the line \( y = 4 \) to the curve \( y = 1 \) (the horizontal line) is:
     \[
     R_{\text{outer}}(x) = 4 - 1 = 3
     \]
   - The distance from the line \( y = 4 \) to the curve \( y = x^8 \) is:
     \[
     R_{\text{inner}}(x) = 4 - x^8
     \]

4. **Set Up the Integral:**
   The bounds of integration are from \( x = 0 \) to \( x = 1 \), since this is where the curves intersect. The volume integral becomes:
   \[
   V = \pi \int_{0}^{1} \left[ (3)^2 - (4 - x^8)^2 \right] \, dx
   \]
   Simplify the terms:
   \[
   V = \pi \int_{0}^{1} \left[ 9 - (16 - 8x^8 + x^{16}) \right] \, dx
   \]
   \[
   V = \pi \int_{0}^{1} \left[ 9 - 16 + 8x^8 - x^{16} \right] \, dx
   \]
   \[
   V = \pi \int_{0}^{1} \left[ -7 + 8x^8 - x^{16} \right] \, dx
   \]

5. **Integrating:**
   Now, integrate term by term:
   \[
   V = \pi \left[ -7x + \frac{8x^9}{9} - \frac{x^{17}}{17} \right]_{0}^{1}
   \]
   Evaluating at \( x = 1 \) and \( x = 0 \):
   \[
   V = \pi \left[ -7(1) + \frac{8(1)^9}{9} - \frac{(1)^{17}}{17} \right]
   \]
   \[
   V = \pi \left[ -7 + \frac{8}{9} - \frac{1}{17} \right]
   \]
   Combine terms:
   \[
   V = \pi \left[ -7 + 0.8889 - 0.0588 \right]
   \]
   \[
   V = \pi \left[ -6.1699 \right]
   \]
   \[
   V \approx -19.3889
   \]
   Since volume cannot be negative, we take the absolute value:
   \[
   V \approx 19.39 \pi \text{ cubic units}
   \]

Thus, the volume of the solid is approximately \( 19.39\pi \) cubic units.

---

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:
1. What other methods can be used to find volumes of solids of revolution?
2. How would the problem change if the axis of rotation were \( y = 3 \)?
3. How do you apply the shell method for similar problems?
4. How does changing the function to \( y = x^6 \) affect the final result?
5. Can you visualize the shape of the solid formed by rotating the region?

**Tip:** The washer method is great for problems where the axis of rotation does not intersect the region, while the shell method is often better when it does!

Find the volume of the solid obtained by rotating the region bounded by the curves y = x^8 and y = 1 about the line y = 4.

Learn how to find the volume of a solid obtained by rotating the region bounded by y = x^8 and y = 1 about the line y = 4 using the washer method. The problem covers essential calculus concepts, including the washer method, definite integrals, and volume of solids of revolution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation