Let A be the region bounded by y = x^3, y=8, and x=0. Set up and integral for the volume of the solid obtained by revolving A about the x-axis.

To find the volume of the solid obtained by revolving the region \( A \) around the x-axis, we can use the **Disk Method**. The Disk Method involves integrating the area of circular disks perpendicular to the axis of rotation.

### Step 1: Identify the curves and boundaries
- The curves given are \( y = x^3 \), \( y = 8 \), and \( x = 0 \).
- The region \( A \) is bounded vertically by \( y = x^3 \) and \( y = 8 \), and horizontally by \( x = 0 \).

### Step 2: Set up the integral
To use the Disk Method, we revolve the region around the x-axis. The radius of each disk is the vertical distance from the x-axis to the curve \( y = x^3 \) and \( y = 8 \).

The volume \( V \) of the solid is given by:
\[
V = \pi \int_{a}^{b} \left[R_{\text{outer}}(x)^2 - R_{\text{inner}}(x)^2\right] \, dx
\]
where:
- \( R_{\text{outer}}(x) = 8 \) (since the outer curve is \( y = 8 \))
- \( R_{\text{inner}}(x) = x^3 \) (since the inner curve is \( y = x^3 \))
- The limits of integration are determined by the points where the curves intersect, \( x = 0 \) and \( x = 2 \) (since \( y = 8 \) intersects \( y = x^3 \) when \( x^3 = 8 \), giving \( x = 2 \)).

### Step 3: Write the integral
The integral for the volume is:
\[
V = \pi \int_{0}^{2} \left[8^2 - (x^3)^2\right] \, dx
\]

Simplifying the integrand:
\[
V = \pi \int_{0}^{2} \left[64 - x^6\right] \, dx
\]

This integral represents the volume of the solid obtained by revolving the region \( A \) about the x-axis.

Would you like to see the steps for evaluating this integral?

---

**Related Questions:**

1. How do you find the volume of a solid using the Washer Method?
2. What changes if the region is revolved around the y-axis instead?
3. How do you determine the limits of integration for solids of revolution?
4. How does the volume calculation differ if the curves are not simple functions?
5. Can the Shell Method be used to find the volume of the solid in this problem?

**Tip:** Always double-check the limits of integration when setting up the integral for solids of revolution, especially when the region is bounded by different curves.

Let A be the region bounded by y = x^3, y=8, and x=0. Set up an integral for the volume of the solid obtained by revolving A about the x-axis.

Learn how to set up and solve an integral to find the volume of a solid obtained by revolving the region bounded by y = x^3, y = 8, and x = 0 about the x-axis. This problem uses the Disk Method, suitable for students in Grades 11-12 and college-level calculus courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation