Introduction to Finding the Volume Formed by Revolving an Area Around a Line

In this detailed guide, we will explore the process of determining the volume created when a specific area is rotated around a given line. We will be diving into the example of the volume generated by revolving the region bounded by the curve y 4x - x^2 and the line x y around the line x 3. We will use the method of cylindrical shells to solve this problem, a technique that is invaluable in solving similar problems in calculus.

Step 1: Determining the Points of Intersection

The first step in our journey is to find the points of intersection between the curve and the line. We do this by setting the equations y 4x - x^2 and y x equal to each other and solving for x.

```python from sympy import symbols, Eq, solve x symbols('x') equation Eq(4*x - x**2, x) solutions solve(equation, x) solutions ```

We can solve the equation by rearranging it:

```python from sympy import simplify equation_rearranged simplify(4*x - x**2 - x) solutions_rearranged solve(equation_rearranged, x) solutions_rearranged ```

The solutions are:

```python [x, 3] ```

This means that the points of intersection are at x 0 and x 3.

Step 2: Setting Up the Volume Integral

Now that we know the points of intersection, we can proceed to set up the volume integral using the method of cylindrical shells. The volume ( V ) is given by the formula:

[ V 2pi int_{a}^{b} text{radius} times text{height} , dx ]

In this case, the radius is the distance from the line x 3 to the curve y x, which is 3 - y. The height is the difference between the curve and the line, which is:

[ 3x - x^2 - x 2x - x^2 ]

Thus, the integral becomes:

[ V 2pi int_{0}^{3} (3 - y) (2x - x^2) , dx ]

Step 3: Adjusting the Integral

Since we are revolving around the line x 3 and we have expressed the height in terms of x, we need to adjust the integral. The radius is actually 3 - x. Thus, the height remains the same, and the integral is:

```python from sympy import integrate, pi integral integrate(2*pi * (3 - x) * (3*x - x**2), (x, 0, 3)) () ```

This integral is:

[ V 2pi int_{0}^{3} (3 - x) (3x - x^2) , dx ]

Step 4: Evaluating the Integral

Let's now evaluate the integral step-by-step:

The integral simplifies to:

[ V 2pi int_{0}^{3} (9x - 6x^2 - x^3) , dx ]

We can now compute the integral:

```python # Define the integrand integrand 9*x - 6*x**2 - x**3 # Compute the definite integral integral_result integrate(integrand, (x, 0, 3)) integral_() ```

The definite integral evaluates to:

[ V 2pi left[ frac{9x^2}{2} - 2x^3 - frac{x^4}{4} right]_0^3 ]

Now we evaluate this at the bounds x 0 and x 3:

[ V 2pi left( frac{9(3^2)}{2} - 2(3^3) - frac{3^4}{4} right) ]

This simplifies to:

[ V 2pi left( frac{81}{2} - 54 - frac{81}{4} right) ]

Converting 54 to quarters:

[ 54 frac{216}{4} ]

So the expression becomes:

[ V 2pi left( frac{162}{4} - frac{216}{4} - frac{81}{4} right) ]

Combining the terms:

[ V 2pi left( frac{27}{4} right) frac{54pi}{4} frac{27pi}{2} ]

Hence, the volume of the solid formed is:

```python 27 * pi / 2 ```

Thus, the final answer is:

The volume of the solid formed is V frac{27pi}{2}.

Conclusion

This detailed exploration has allowed us to understand the method of cylindrical shells and how it can be used to solve problems involving the volume formed by revolving an area around a line. The techniques discussed here are not only applicable to this particular problem but can be a powerful tool for solving similar calculus problems.