How to Find the Volume of a Solid Generated by Revolving Given Curves Around a Line

In this detailed guide, we will explore the process of finding the volume of a solid generated by revolving the area bounded by given curves around a line. Specifically, we will address the problem:

Problem Statement

Find the volume of the solid generated by revolving the area bounded by the curves (xy 4), (x 2), and (y 4) around the line (y 4).

Step-by-Step Solution

Step 1: Identify the Area Bounded by the Curves

The equation (xy 4) can be rewritten as (y frac{4}{x}). This hyperbola intersects the line (y 4) at (x 1) since (4 frac{4}{1}). The vertical line (x 2) intersects the hyperbola at (y frac{4}{2} 2). The line (y 4) serves as the axis of rotation.

Step 2: Determine the Region of Integration

The area of interest is bounded by:

The hyperbola (y frac{4}{x}) from (x 1) to (x 2) The vertical line (x 2) The horizontal line (y 4)

Step 3: Set Up the Volume Integral

For the volume integral using the washer method, the volume (V) is given by the formula:

(V pi int_{a}^{b} (R^2 - r^2) dx)

where:

(R) is the outer radius distance from (y 4) (r) is the inner radius distance from (y 4)

Step 4: Determine Radii

Outer Radius (R): The distance from (y 4) to the line (y 4) is (0) since it is the same line.

Inner Radius (r): The distance from (y 4) to the hyperbola (y frac{4}{x}) is (4 - frac{4}{x}).

Step 5: Set Up the Integral

The volume integral becomes:

(V pi int_{1}^{2} left(0^2 - left(4 - frac{4}{x}right)^2right) dx)

This simplifies to:

(V pi int_{1}^{2} left(4 - frac{4}{x} - frac{16}{x^2}right) dx)

Step 6: Calculate the Integral

First, expand the integrand:

(left(4 - frac{4}{x} - frac{16}{x^2}right) 4 - frac{4}{x} - frac{16}{x^2})

Now the volume integral is:

(V pi int_{1}^{2} left(4 - frac{32}{x} frac{16}{x^2}right) dx)

Step 7: Integrate Each Term

Integral of (4): (int 4 dx 4x)

Integral of (-frac{32}{x}): (int -frac{32}{x} dx -32 ln x)

Integral of (-frac{16}{x^2}): (int -frac{16}{x^2} dx frac{16}{x})

Step 8: Combine and Evaluate the Integral

Putting it all together:

(V pi left[ 4x - 32 ln x frac{16}{x} right]_{1}^{2})

Evaluating at the bounds:

At (x 2): (16 - 32 ln 2 - 8 24 - 32 ln 2)

At (x 1): (4 - 0 16 20)

Step 9: Final Volume Calculation

Thus, the volume is:

(V pi left[ 24 - 32 ln 2 - 0 right] pi (24 - 32 ln 2))

Conclusion

The volume of the solid generated by revolving the given area around the line (y 4) is:

(V pi (24 - 32 ln 2))

By following these steps and applying the appropriate integration techniques, you can successfully calculate the volume of a solid generated by revolving a given area around a specified line.