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

Introduction

In calculus, the volume of a solid obtained by rotating a region about an axis is a fundamental concept. The method of disks (or washers) is used to calculate the volume of such solids. However, when the region is not bounded at x=0, the calculation becomes more complex. In this article, we will discuss how to find the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0.

Problem Statement

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.

Understanding the Region

To solve this problem, we need to understand the region that is being rotated. The region is bounded by the parabola $y = 2x^2$ and the lines $x = 2$ and $x = 4$. The region is not bounded at x=0, which means that the region extends from x=2 to x=4.

Method of Disks

The method of disks is used to calculate the volume of a solid obtained by rotating a region about an axis. The formula for the volume of a solid obtained by rotating a region about the x-axis is given by:

$$V = \pi \int_{a}^{b} [f(x)]^2 dx$$

where $f(x)$ is the function that defines the region, and $a$ and $b$ are the limits of integration.

Calculating the Volume

In this case, the function that defines the region is $y = 2x^2$. The limits of integration are $x = 2$ and $x = 4$. Therefore, the volume of the solid obtained by rotating the region about the x-axis is given by:

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

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

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

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

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

$$V = \pi \left(\frac{4096}{5} - \frac{128}{5}\right)$$

$$V = \pi \left(\frac{3968}{5}\right)$$

$$V = \frac{3968\pi}{5}$$

Conclusion

In this article, we discussed how to find the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0. We used the method of disks to calculate 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. The volume of the solid is given by $\frac{3968\pi}{5}$.

Additional Information

The problem statement 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. However, the solution we obtained is not the same as the one you mentioned in the problem statement. The solution you mentioned is $\int_{8}^{32} 2\pi y \...$. This is because the region is not bounded at x=0, and the method of disks needs to be adjusted accordingly.

Adjusting the Method of Disks

When the region is not bounded at x=0, the method of disks needs to be adjusted. The formula for the volume of a solid obtained by rotating a region about the x-axis is given by:

$$V = \pi \int_{a}^{b} [f(x)]^2 dx$$

However, in this case, the region is not bounded at x=0, and the function $f(x)$ is not defined at x=0. Therefore, we need to adjust the formula for the volume of a solid obtained by rotating a region about the x-axis.

Adjusted Formula

The adjusted formula for the volume of a solid obtained by rotating a region about the x-axis is given by:

$$V = \pi \int_{a}^{b} [f(x)]^2 dx + \pi \int_{0}^{a} [f(x)]^2 dx$$

where $a$ and $b$ are the limits of integration.

Applying the Adjusted Formula

In this case, the function that defines the region is $y = 2x^2$. The limits of integration are $x = 2$ and $x = 4$. Therefore, the volume of the solid obtained by rotating the region about the x-axis is given by:

$$V = \pi \int_{2}^{4} (2x^2)^2 dx + \pi \int_{0}^{2} (2x^2)^2 dx$$

$$V = \pi \int_{2}^{4} 4x^4 dx + \pi \int_{0}^{2} 4x^4 dx$$

$$V = \pi \left[\frac{4x^5}{5}\right]_{2}^{4} + \pi \left[\frac{4x^5}{5}\right]_{0}^{2}$$

$$V = \pi \left(\frac{4(4)^5}{5} - \frac{4(2)^5}{5}\right) + \pi \left(\frac{4(2)^5}{5} - \frac{4(0)^5}{5}\right)$$

$$V = \pi \left(\frac{4(1024)}{5} - \frac{4(32)}{5}\right) + \pi \left(\frac{4(32)}{5} - 0\right)$$

$$V = \pi \left(\frac{4096}{5} - \frac{128}{5}\right) + \pi \left(\frac{128}{5}\right)$$

$$V = \pi \left(\frac{3968}{5}\right) + \pi \left(\frac{128}{5}\right)$$

$$V = \pi \left(\frac{4096}{5}\right)$$

Conclusion

In this article, we discussed how to find the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0. We used the adjusted formula for the volume of a solid obtained by rotating a region about the x-axis to calculate 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. The volume of the solid is given by $\frac{3968\pi}{5}$.

Final Answer

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Introduction

In our previous article, we discussed how to find the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0. We used the method of disks to calculate 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. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the method of disks?

A: The method of disks is a technique used to calculate the volume of a solid obtained by rotating a region about an axis. The formula for the volume of a solid obtained by rotating a region about the x-axis is given by:

$$V = \pi \int_{a}^{b} [f(x)]^2 dx$$

where $f(x)$ is the function that defines the region, and $a$ and $b$ are the limits of integration.

Q: What is the adjusted formula for the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0?

A: The adjusted formula for the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0 is given by:

$$V = \pi \int_{a}^{b} [f(x)]^2 dx + \pi \int_{0}^{a} [f(x)]^2 dx$$

where $a$ and $b$ are the limits of integration.

Q: How do I apply the adjusted formula to calculate the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0?

A: To apply the adjusted formula, you need to follow these steps:

1. Identify the function that defines the region.
2. Identify the limits of integration.
3. Calculate the integral of the function squared from the lower limit of integration to the upper limit of integration.
4. Calculate the integral of the function squared from 0 to the lower limit of integration.
5. Add the results of steps 3 and 4 to get the volume of the solid.

Q: What is 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?

A: 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 is given by:

$$V = \frac{3968\pi}{5}$$

Q: How do I calculate the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0 using the method of disks?

A: To calculate the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0 using the method of disks, you need to follow these steps:

1. Identify the function that defines the region.
2. Identify the limits of integration.
3. Calculate the integral of the function squared from the lower limit of integration to the upper limit of integration.
4. Add the result of step 3 to the integral of the function squared from 0 to the lower limit of integration.

Q: What are some common mistakes to avoid when calculating the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0?

A: Some common mistakes to avoid when calculating the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0 include:

• Failing to identify the function that defines the region.
• Failing to identify the limits of integration.
• Failing to calculate the integral of the function squared from the lower limit of integration to the upper limit of integration.
• Failing to calculate the integral of the function squared from 0 to the lower limit of integration.
• Failing to add the results of the two integrals to get the volume of the solid.

Conclusion

In this article, we answered some frequently asked questions related to the topic of finding the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0. We discussed the method of disks, the adjusted formula for the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0, and how to apply the adjusted formula to calculate the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0. We also discussed some common mistakes to avoid when calculating the volume of a solid obtained by rotating a region about the x-axis when it is not bounded at x=0.