Using the Chop/Disk/Washer technique, setup an integral that represents the volume of the figure generated by the shaded area by y=x^2+2, y=6 reolved around x axis.

To find the volume of the solid generated by revolving the region between \( y = x^2 + 2 \) and \( y = 6 \) around the x-axis using the Disk/Washer method, we first outline the necessary steps:

### Step 1: Identify the curves and the limits of integration
The two curves involved are:

- \( y = x^2 + 2 \) (a parabola opening upwards)
- \( y = 6 \) (a horizontal line)

We need to determine where the two curves intersect by solving for \( x \) where \( y = x^2 + 2 \) and \( y = 6 \):

\[
x^2 + 2 = 6 \implies x^2 = 4 \implies x = \pm 2
\]

So, the region of interest is between \( x = -2 \) and \( x = 2 \).

### Step 2: Set up the integral using the Washer Method
The volume \( V \) of the solid formed by revolving the region around the x-axis is given by the formula:

\[
V = \pi \int_{a}^{b} \left[ R_{\text{outer}}^2(x) - R_{\text{inner}}^2(x) \right] dx
\]

- The **outer radius** \( R_{\text{outer}}(x) \) is the distance from the x-axis to the upper curve \( y = 6 \), which is constant: \( R_{\text{outer}}(x) = 6 \).
- The **inner radius** \( R_{\text{inner}}(x) \) is the distance from the x-axis to the lower curve \( y = x^2 + 2 \), so \( R_{\text{inner}}(x) = x^2 + 2 \).

### Step 3: Write the integral expression
Now, set up the integral:

\[
V = \pi \int_{-2}^{2} \left[ 6^2 - (x^2 + 2)^2 \right] dx
\]

This simplifies to:

\[
V = \pi \int_{-2}^{2} \left[ 36 - (x^4 + 4x^2 + 4) \right] dx
\]

Finally:

\[
V = \pi \int_{-2}^{2} \left[ 36 - x^4 - 4x^2 - 4 \right] dx = \pi \int_{-2}^{2} \left[ 32 - x^4 - 4x^2 \right] dx
\]

This is the integral that represents the volume of the solid.

Would you like me to solve this integral, or do you need further clarification?

---

Here are 5 related questions for further exploration:
1. How would the setup change if the region was revolved around the line \( y = 2 \)?
2. What would happen if the same region were revolved around the y-axis instead?
3. How does the Washer Method differ from the Shell Method for this problem?
4. Can you calculate the volume using cylindrical shells instead of washers?
5. How would the volume change if the region between \( y = 6 \) and \( y = 3 \) were revolved?

**Tip:** Always sketch the region and axis of revolution to visualize the problem before setting up integrals.

Using the Chop/Disk/Washer technique, setup an integral that represents the volume of the figure generated by the shaded area by y=x^2+2, y=6 resolved around x-axis.

Learn how to set up the integral to calculate the volume of the solid formed by revolving the region between y=x^2+2 and y=6 around the x-axis using the Washer Method. This step-by-step solution includes calculus concepts, definite integrals, and the application of the washer method. Suitable for students in Grades 11-12 or early university.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation