Volume Of Rotating Solid About The X-axis Not Bounded At X=0 Using Shell Method

Introduction

In calculus, the shell method is a technique used to find the volume of a solid of revolution. It involves integrating the area of cylindrical shells with respect to the axis of rotation. In this article, we will use the shell method to find the volume of a solid obtained by rotating a region bounded by a parabola and two lines about the x-axis.

Problem Statement

The problem asks us to find the volume of the solid obtained by rotating the region bounded by the parabola $y = 2x^2$ and the lines $x = 2$, $x = 4$, and $y = 0$ about the x-axis using the shell method.

Understanding the Shell Method

The shell method involves integrating the area of cylindrical shells with respect to the axis of rotation. To apply the shell method, we need to:

1. Identify the axis of rotation: In this case, the axis of rotation is the x-axis.
2. Find the height of the shell: The height of the shell is the distance between the curve and the x-axis, which is given by the function $y = 2x^2$.
3. Find the radius of the shell: The radius of the shell is the distance between the axis of rotation and the edge of the shell, which is given by the x-coordinate of the shell.
4. Integrate the area of the shell: The area of the shell is given by the formula $2\pi rh$, where $r$ is the radius of the shell and $h$ is the height of the shell.

Applying the Shell Method

To apply the shell method, we need to integrate the area of the shell with respect to the x-axis. The limits of integration are from $x = 2$ to $x = 4$.

The height of the shell is given by the function $y = 2x^2$, and the radius of the shell is given by the x-coordinate of the shell.

The area of the shell is given by the formula $2\pi rh$, where $r$ is the radius of the shell and $h$ is the height of the shell.

The integral to be evaluated is:

$$V = \int_{2}^{4} 2\pi x(2x^2) dx$$

Evaluating the Integral

To evaluate the integral, we need to follow the order of operations:

1. Multiply the terms: $2\pi x(2x^2) = 4\pi x^3$
2. Integrate the term: $\int 4\pi x^3 dx = \frac{4\pi x^4}{4} = \pi x^4$
3. Apply the limits of integration: $\pi x^4 \Big|_{2}^{4} = \pi (4^4) - \pi (2^4) = \pi (256) - \pi (16) = 240\pi$

Conclusion

The volume of the solid obtained by rotating the region bounded by the parabola $y = 2x^2$ and the lines $x = 2$, $x = 4$, and $y = 0$ about the x-axis is $240\pi$ cubic units.

Example Use Cases

The shell method can be used to find the volume of a variety of solids of revolution, including:

• Cones: The shell method can be used to find the volume of a cone by rotating a right triangle about its base.
• Spheres: The shell method can be used to find the volume of a sphere by rotating a circle about its diameter.
• Toruses: The shell method can be used to find the volume of a torus by rotating a circle about a line that is not its diameter.

Limitations of the Shell Method

The shell method has several limitations, including:

• Difficulty in applying the method: The shell method can be difficult to apply in certain cases, such as when the axis of rotation is not the x-axis.
• Difficulty in evaluating the integral: The integral to be evaluated in the shell method can be difficult to evaluate in certain cases, such as when the function is not a polynomial.

Conclusion

Q&A: Shell Method for Volume of Rotating Solid

Q: What is the shell method?

A: The shell method is a technique used to find the volume of a solid of revolution. It involves integrating the area of cylindrical shells with respect to the axis of rotation.

Q: What are the steps to apply the shell method?

A: The steps to apply the shell method are:

1. Identify the axis of rotation: In this case, the axis of rotation is the x-axis.
2. Find the height of the shell: The height of the shell is the distance between the curve and the x-axis, which is given by the function $y = 2x^2$.
3. Find the radius of the shell: The radius of the shell is the distance between the axis of rotation and the edge of the shell, which is given by the x-coordinate of the shell.
4. Integrate the area of the shell: The area of the shell is given by the formula $2\pi rh$, where $r$ is the radius of the shell and $h$ is the height of the shell.

Q: How do I find the height of the shell?

A: The height of the shell is given by the function $y = 2x^2$. This function represents the distance between the curve and the x-axis.

Q: How do I find the radius of the shell?

A: The radius of the shell is given by the x-coordinate of the shell. This is the distance between the axis of rotation and the edge of the shell.

Q: What is the formula for the area of the shell?

A: The area of the shell is given by the formula $2\pi rh$, where $r$ is the radius of the shell and $h$ is the height of the shell.

Q: How do I integrate the area of the shell?

A: To integrate the area of the shell, you need to follow the order of operations:

1. Multiply the terms: $2\pi x(2x^2) = 4\pi x^3$
2. Integrate the term: $\int 4\pi x^3 dx = \frac{4\pi x^4}{4} = \pi x^4$
3. Apply the limits of integration: $\pi x^4 \Big|_{2}^{4} = \pi (4^4) - \pi (2^4) = \pi (256) - \pi (16) = 240\pi$

Q: What are some examples of solids of revolution that can be found using the shell method?

A: Some examples of solids of revolution that can be found using the shell method include:

• Cones: The shell method can be used to find the volume of a cone by rotating a right triangle about its base.
• Spheres: The shell method can be used to find the volume of a sphere by rotating a circle about its diameter.
• Toruses: The shell method can be used to find the volume of a torus by rotating a circle about a line that is not its diameter.

Q: What are some limitations of the shell method?

A: Some limitations of the shell method include:

• Difficulty in applying the method: The shell method can be difficult to apply in certain cases, such as when the axis of rotation is not the x-axis.
• Difficulty in evaluating the integral: The integral to be evaluated in the shell method can be difficult to evaluate in certain cases, such as when the function is not a polynomial.

Q: How do I choose between the shell method and the disk method?

A: The choice between the shell method and the disk method depends on the specific problem and the axis of rotation. If the axis of rotation is the x-axis, the shell method may be more convenient. If the axis of rotation is not the x-axis, the disk method may be more convenient.

Conclusion

In conclusion, the shell method is a powerful technique used to find the volume of a solid of revolution. It involves integrating the area of cylindrical shells with respect to the axis of rotation. The shell method can be used to find the volume of a variety of solids of revolution, including cones, spheres, and toruses. However, the shell method has several limitations, including difficulty in applying the method and difficulty in evaluating the integral.