Find the volume *V* of the solid obtained by rotating the region bounded by the given curves about the specified line.

*y* =

*x* − 1

,  *y* = 0,  *x* = 8;    about the *x*-axis

To find the volume \( V \) of the solid obtained by rotating the region bounded by the curves about the \( x \)-axis, we can apply the **Disk Method**.

### Step-by-Step Approach:

1. **Identify the curves and region:**
   - The region is bounded by:
     - \( y = x - 1 \)
     - \( y = 0 \) (the x-axis)
     - \( x = 8 \)

2. **Set up the Disk Method formula:**

   The volume of a solid obtained by rotating a region about the \( x \)-axis is given by:
   \[
   V = \pi \int_{a}^{b} [f(x)]^2 \, dx
   \]
   where \( f(x) \) is the function describing the curve being rotated, and \( [a, b] \) are the limits of integration along the \( x \)-axis.

3. **Set up the integral:**

   The curve \( y = x - 1 \) will be rotated about the \( x \)-axis. This gives the radius of each disk as \( f(x) = x - 1 \). The limits of integration are from \( x = 1 \) (where \( y = 0 \)) to \( x = 8 \) (as given in the problem).

   So, the integral becomes:
   \[
   V = \pi \int_{1}^{8} (x - 1)^2 \, dx
   \]

4. **Expand the integrand:**

   First, expand \( (x - 1)^2 \):
   \[
   (x - 1)^2 = x^2 - 2x + 1
   \]
   Now, the integral becomes:
   \[
   V = \pi \int_{1}^{8} (x^2 - 2x + 1) \, dx
   \]

5. **Integrate:**

   Now, integrate each term:
   \[
   V = \pi \left[ \frac{x^3}{3} - x^2 + x \right]_{1}^{8}
   \]

6. **Evaluate the definite integral:**

   First, evaluate at \( x = 8 \):
   \[
   \left( \frac{8^3}{3} - 8^2 + 8 \right) = \left( \frac{512}{3} - 64 + 8 \right) = \frac{512}{3} - 56 = \frac{512 - 168}{3} = \frac{344}{3}
   \]

   Next, evaluate at \( x = 1 \):
   \[
   \left( \frac{1^3}{3} - 1^2 + 1 \right) = \left( \frac{1}{3} - 1 + 1 \right) = \frac{1}{3}
   \]

   Now, subtract the two values:
   \[
   V = \pi \left( \frac{344}{3} - \frac{1}{3} \right) = \pi \left( \frac{343}{3} \right)
   \]

7. **Final answer:**

   The volume of the solid is:
   \[
   V = \frac{343\pi}{3}
   \]

### Do you need further details or clarifications?

---

Here are 5 related questions to expand on the topic:
1. How does the Disk Method differ from the Shell Method?
2. What would the volume be if the region were rotated about the line \( y = 1 \)?
3. Can you solve the same problem using cylindrical shells instead of disks?
4. How would the volume change if the curve were rotated about the \( y \)-axis?
5. How do you find the volume of a solid with a more complex boundary, such as \( y = x^2 \)?

**Tip:** Always check the limits of integration carefully when setting up an integral for volumes of rotation.

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x - 1, y = 0, x = 8; about the x-axis.

Learn how to find the volume of the solid obtained by rotating the region bounded by y = x - 1, y = 0, and x = 8 about the x-axis using the Disk Method. This problem covers integral calculus concepts and is suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation