ind the volume of the solid generated by revolving the shaded region about the​ x-axis.
The curve is y equals StartFraction 14 Over StartRoot 14 x minus x squared EndRoot EndFraction
​; x 1 equals 1.75​, x 2 equals 12.25.

To find the volume of the solid generated by revolving the shaded region about the x-axis, we use the **Disk Method**. The formula for the volume of revolution about the x-axis is given by:

\[
V = \pi \int_{x_1}^{x_2} [f(x)]^2 \, dx
\]

Where:
- \( f(x) \) is the function of the curve.
- \( x_1 \) and \( x_2 \) are the limits of integration.

Given the function:

\[
y = \frac{14}{\sqrt{14x - x^2}}
\]

and \( x_1 = 1.75 \) and \( x_2 = 12.25 \), we can now set up the integral for the volume.

### Step 1: Set up the integral
We will square the function \( y = f(x) \), and substitute it into the volume formula:

\[
V = \pi \int_{1.75}^{12.25} \left( \frac{14}{\sqrt{14x - x^2}} \right)^2 \, dx
\]

### Step 2: Simplify the expression
Simplifying \( \left( \frac{14}{\sqrt{14x - x^2}} \right)^2 \):

\[
\left( \frac{14}{\sqrt{14x - x^2}} \right)^2 = \frac{196}{14x - x^2}
\]

So, the integral becomes:

\[
V = \pi \int_{1.75}^{12.25} \frac{196}{14x - x^2} \, dx
\]

### Step 3: Solve the integral
To calculate this integral, we can either attempt a substitution or solve it numerically. This type of integral is more easily solved numerically using integration tools.

Let me calculate this integral for you.

The volume of the solid generated by revolving the shaded region about the x-axis is approximately \( 171.17 \) cubic units.

Do you want more details or have any questions?

Here are some related questions you might find interesting:
1. How would the volume change if the region were rotated about the y-axis instead?
2. What happens to the volume if the bounds of integration change, say from \( x_1 = 2 \) to \( x_2 = 10 \)?
3. Can we use a different method, like the Shell Method, to find the volume? How would that work?
4. What would happen to the volume if we changed the function to a slightly different curve?
5. How would we compute the surface area of the solid instead of the volume?

**Tip:** When calculating volumes of revolution, always ensure you are using the correct axis of rotation and method (Disk or Shell).

Find the volume of the solid generated by revolving the shaded region about the​ x-axis. The curve is y = 14 / sqrt(14x - x^2), with x1 = 1.75 and x2 = 12.25.

Learn how to calculate the volume of a solid generated by revolving the curve y = 14 / sqrt(14x - x^2) around the x-axis using the Disk Method. This detailed step-by-step solution includes setting up and solving the definite integral, with bounds x1 = 1.75 and x2 = 12.25. Suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation