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

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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 discuss how to find the volume of a solid obtained by rotating a region about the x-axis using the shell method. We will also provide a step-by-step solution to a specific problem.

The Shell Method

The shell method is based on the idea of dividing the solid into thin cylindrical shells. Each shell has a radius equal to the distance from the axis of rotation to the edge of the solid, and a height equal to the distance from the axis of rotation to the top of the solid. The volume of each shell is then calculated as the product of its radius and height, and the sum of the volumes of all the shells is integrated with respect to the axis of rotation.

Mathematical Formulation

Let's consider a solid obtained by rotating a region about the x-axis. The region is bounded by a curve y = f(x) and the lines x = a and x = b. The volume of the solid can be calculated using the shell method as follows:

V = ∫[a,b] 2πrh dx

where r is the radius of the shell, h is the height of the shell, and dx is the thickness of the shell.

Step-by-Step Solution

Now, let's consider the specific problem of finding 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 using the shell method.

Step 1: Define the Region

The region is bounded by the curve y = 2x^2 and the lines x = 2, x = 4, and y = 0. We need to find the volume of the solid obtained by rotating this region about the x-axis.

Step 2: Determine the Limits of Integration

The limits of integration are the x-coordinates of the lines x = 2 and x = 4, which are 2 and 4, respectively.

Step 3: Calculate the Radius and Height of Each Shell

The radius of each shell is equal to the distance from the x-axis to the edge of the solid, which is x. The height of each shell is equal to the distance from the x-axis to the top of the solid, which is 2x^2.

Step 4: Calculate the Volume of Each Shell

The volume of each shell is calculated as the product of its radius and height, which is 2Ï€x(2x^2) = 4Ï€x^3.

Step 5: Integrate the Volumes of All the Shells

The volume of the solid is calculated by integrating the volumes of all the shells with respect to the x-axis. This is done by evaluating the integral:

V = ∫[2,4] 4πx^3 dx

Step 6: Evaluate the Integral

To evaluate the integral, we can use the power rule of integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C. In this case, n = 3, so we have:

V = ∫[2,4] 4πx^3 dx = 4π ∫[2,4] x^3 dx = 4π [(x^4)/4] from 2 to 4 = 4π [(4^4)/4 - (2^4)/4] = 4π [(256)/4 - (16)/4] = 4π [64 - 4] = 4π [60] = 240π

Step 7: Simplify the Answer

The final answer is 240Ï€ cubic units.

Conclusion

In this article, we discussed how to find the volume of a solid obtained by rotating a region about the x-axis using the shell method. We also provided a step-by-step solution to a specific problem. The shell method is a powerful technique for finding the volume of solids of revolution, and it is widely used in calculus and engineering applications.

References

  • [1] "Calculus: Early Transcendentals" by James Stewart
  • [2] "Calculus: Single Variable" by Michael Spivak
  • [3] "Shell Method" by Wolfram MathWorld

Additional Resources

  • [1] "Shell Method" by Khan Academy
  • [2] "Shell Method" by MIT OpenCourseWare
  • [3] "Shell Method" by Wolfram Alpha
    Volume of Rotating Solid about the X-Axis Not Bounded at X=0 Using Shell Method: Q&A ====================================================================================

Introduction

In our previous article, we discussed how to find the volume of a solid obtained by rotating a region about the x-axis using the shell method. We also provided a step-by-step solution to a specific problem. In this article, we will answer some frequently asked questions related to the shell method and volume of solids of revolution.

Q&A

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 advantages of the shell method?

A: The shell method has several advantages, including:

  • It is easy to apply to solids of revolution that are not bounded at x=0.
  • It is useful for finding the volume of solids with complex shapes.
  • It can be used to find the volume of solids with varying cross-sectional areas.

Q: What are the disadvantages of the shell method?

A: The shell method has several disadvantages, including:

  • It can be difficult to apply to solids of revolution that are bounded at x=0.
  • It requires a good understanding of calculus and integration.
  • It can be time-consuming to apply to complex solids.

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 shape of the solid and the axis of rotation. If the solid is not bounded at x=0, the shell method is usually the best choice. If the solid is bounded at x=0, the disk method may be more suitable.

Q: What are some common mistakes to avoid when using the shell method?

A: Some common mistakes to avoid when using the shell method include:

  • Failing to define the region of integration correctly.
  • Failing to calculate the radius and height of each shell correctly.
  • Failing to integrate the volumes of all the shells correctly.

Q: How do I apply the shell method to a solid with a varying cross-sectional area?

A: To apply the shell method to a solid with a varying cross-sectional area, you need to calculate the area of each shell using the formula A = 2Ï€rh, where r is the radius of the shell and h is the height of the shell. You then integrate the areas of all the shells with respect to the axis of rotation.

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 need to calculate the area of each shell using the formula A = 2Ï€rh, where r is the radius of the shell and h is the height of the shell.

Q: How do I apply the shell method to a solid with a complex shape?

A: To apply the shell method to a solid with a complex shape, you need to break down the solid into simpler shapes and calculate the volume of each shape using the shell method. You then add up the volumes of all the shapes to find the total volume of the solid.

Conclusion

In this article, we answered some frequently asked questions related to the shell method and volume of solids of revolution. We hope that this article has been helpful in clarifying some of the concepts and techniques involved in the shell method.

References

  • [1] "Calculus: Early Transcendentals" by James Stewart
  • [2] "Calculus: Single Variable" by Michael Spivak
  • [3] "Shell Method" by Wolfram MathWorld

Additional Resources

  • [1] "Shell Method" by Khan Academy
  • [2] "Shell Method" by MIT OpenCourseWare
  • [3] "Shell Method" by Wolfram Alpha