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

Feb 28, 2025 by ADMIN 80 views

Introduction

The shell method is a technique used in calculus to find the volume of a solid of revolution. It involves integrating the area of thin 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 the region bounded by $y = 2x^2$ and the lines $x = 2$ , $x = 4$ and $y = 0$ about the x-axis.

Understanding the Problem

The problem asks us to find the volume of the solid obtained by rotating the region bounded by $y = 2x^2$ and the lines $x = 2$ , $x = 4$ and $y = 0$ about the x-axis. This means that we need to integrate the area of the region with respect to the x-axis.

The Shell Method

The shell method involves integrating the area of thin cylindrical shells with respect to the axis of rotation. The formula for the shell method is:

V = 2\pi \int_{a}^{b} r(x)h(x) dx

where $V$ is the volume of the solid, $r(x)$ is the radius of the shell, $h(x)$ is the height of the shell, and $a$ and $b$ are the limits of integration.

Finding the Radius and Height of the Shell

To find the radius and height of the shell, we need to analyze the region bounded by $y = 2x^2$ and the lines $x = 2$ , $x = 4$ and $y = 0$ . The radius of the shell is the distance from the x-axis to the curve $y = 2x^2$ , which is given by:

r(x) = 2x^2

The height of the shell is the distance from the curve $y = 2x^2$ to the x-axis, which is given by:

h(x) = 2x^2

Setting Up the Integral

Now that we have found the radius and height of the shell, we can set up the integral using the shell method formula:

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

Evaluating the Integral

To evaluate the integral, we can use the power rule of integration, which states that:

\int x^n dx = \frac{x^{n+1}}{n+1} + C

Using this rule, we can evaluate the integral as follows:

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

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

V = 2\pi \left[\frac{4x^5}{5}\right]_{2}^{4}

V = 2\pi \left[\frac{4(4)^5}{5} - \frac{4(2)^5}{5}\right]

V = 2\pi \left[\frac{4(1024)}{5} - \frac{4(32)}{5}\right]

V = 2\pi \left[\frac{4096}{5} - \frac{128}{5}\right]

V = 2\pi \left[\frac{3968}{5}\right]

V = 2\pi \left[793.6\right]

V = 5003.2\pi

Conclusion

In this article, we used the shell method to find the volume of a solid obtained by rotating the region bounded by $y = 2x^2$ and the lines $x = 2$ , $x = 4$ and $y = 0$ about the x-axis. We found that the volume of the solid is given by:

V = 5003.2\pi

This result shows that the shell method is a powerful tool for finding the volume of solids of revolution.

References

[1] "Calculus: Early Transcendentals" by James Stewart
[2] "Calculus: Single Variable" by Michael Spivak

Additional Information

The shell method is a technique used in calculus to find the volume of a solid of revolution.
The formula for the shell method is:

V = 2\pi \int_{a}^{b} r(x)h(x) dx

where $V$ is the volume of the solid, $r(x)$ is the radius of the shell, $h(x)$ is the height of the shell, and $a$ and $b$ are the limits of integration.

The shell method involves integrating the area of thin cylindrical shells with respect to the axis of rotation.

Frequently Asked Questions

Q: What is the shell method? A: The shell method is a technique used in calculus to find the volume of a solid of revolution.
Q: How do I use the shell method to find the volume of a solid? A: To use the shell method, you need to find the radius and height of the shell, and then integrate the area of the shell with respect to the axis of rotation.
Q: What are the limits of integration for the shell method? A: The limits of integration for the shell method are the limits of the region being rotated.
Frequently Asked Questions about the Shell Method =====================================================

Q: What is the shell method?

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

Q: How do I use the shell method to find the volume of a solid?

A: To use the shell method, you need to follow these steps:

Find the radius and height of the shell: The radius of the shell is the distance from the axis of rotation to the curve being rotated, while the height of the shell is the distance from the curve to the axis of rotation.
Set up the integral: Use the formula for the shell method to set up the integral:

V = 2\pi \int_{a}^{b} r(x)h(x) dx

where $V$ is the volume of the solid, $r(x)$ is the radius of the shell, $h(x)$ is the height of the shell, and $a$ and $b$ are the limits of integration. 3. Evaluate the integral: Use integration techniques to evaluate the integral and find the volume of the solid.

Q: What are the limits of integration for the shell method?

A: The limits of integration for the shell method are the limits of the region being rotated. These limits are typically given as $a$ and $b$ , where $a$ is the lower limit and $b$ is the upper limit.

Q: How do I choose the axis of rotation for the shell method?

A: The axis of rotation is typically chosen to be the axis that is most convenient for the problem. For example, if the region being rotated is a circle, it may be easier to use the shell method with the axis of rotation as the diameter of the circle.

Q: Can I use the shell method to find the volume of a solid with a non-circular cross-section?

A: Yes, you can use the shell method to find the volume of a solid with a non-circular cross-section. However, you will need to use a more general formula for the shell method, which takes into account the shape of the cross-section.

Q: How do I handle the case where the axis of rotation is not perpendicular to the region being rotated?

A: If the axis of rotation is not perpendicular to the region being rotated, you will need to use a more general formula for the shell method, which takes into account the angle between the axis of rotation and the region being rotated.

Q: Can I use the shell method to find the volume of a solid with a hole in it?

A: Yes, you can use the shell method to find the volume of a solid with a hole in it. However, you will need to subtract the volume of the hole from the volume of the solid.

Q: How do I handle the case where the region being rotated is not a simple shape?

A: If the region being rotated is not a simple shape, you may need to use a more general formula for the shell method, which takes into account the shape of the region being rotated.

Q: Can I use the shell method to find the volume of a solid with a complex boundary?

A: Yes, you can use the shell method to find the volume of a solid with a complex boundary. However, you will need to use a more general formula for the shell method, which takes into account the shape of the boundary.

Q: How do I choose the limits of integration for the shell method when the region being rotated is not a simple shape?

A: When the region being rotated is not a simple shape, you will need to choose the limits of integration carefully. You may need to use a more general formula for the shell method, which takes into account the shape of the region being rotated.

Q: Can I use the shell method to find the volume of a solid with a non-rectangular cross-section?

A: Yes, you can use the shell method to find the volume of a solid with a non-rectangular cross-section. However, you will need to use a more general formula for the shell method, which takes into account the shape of the cross-section.

Q: How do I handle the case where the axis of rotation is not parallel to the region being rotated?

A: If the axis of rotation is not parallel to the region being rotated, you will need to use a more general formula for the shell method, which takes into account the angle between the axis of rotation and the region being rotated.

Q: Can I use the shell method to find the volume of a solid with a complex shape?

A: Yes, you can use the shell method to find the volume of a solid with a complex shape. However, you will need to use a more general formula for the shell method, which takes into account the shape of the solid.

Q: How do I choose the limits of integration for the shell method when the region being rotated is a complex shape?

A: When the region being rotated is a complex shape, you will need to choose the limits of integration carefully. You may need to use a more general formula for the shell method, which takes into account the shape of the region being rotated.

Q: Can I use the shell method to find the volume of a solid with a non-uniform density?

A: Yes, you can use the shell method to find the volume of a solid with a non-uniform density. However, you will need to use a more general formula for the shell method, which takes into account the density of the solid.

Q: How do I handle the case where the axis of rotation is not perpendicular to the region being rotated and the region being rotated is not a simple shape?

A: If the axis of rotation is not perpendicular to the region being rotated and the region being rotated is not a simple shape, you will need to use a more general formula for the shell method, which takes into account the angle between the axis of rotation and the region being rotated, as well as the shape of the region being rotated.

Q: Can I use the shell method to find the volume of a solid with a complex boundary and a non-uniform density?

A: Yes, you can use the shell method to find the volume of a solid with a complex boundary and a non-uniform density. However, you will need to use a more general formula for the shell method, which takes into account the shape of the boundary and the density of the solid.

Q: How do I choose the limits of integration for the shell method when the region being rotated is a complex shape and has a non-uniform density?

A: When the region being rotated is a complex shape and has a non-uniform density, you will need to choose the limits of integration carefully. You may need to use a more general formula for the shell method, which takes into account the shape of the region being rotated and the density of the solid.