Finding the Volume of the Solid of Revolution

Oct 13, 2025 by miodeya.com 4 views

Introduction

In mathematics, the concept of volume of solid of revolution is a fundamental topic that deals with finding the volume of a solid formed by rotating a region around an axis. This concept is widely used in various fields such as physics, engineering, and architecture. In this article, we will discuss how to set up definite integrals with respect to $y$ to find the volume of the solid of revolution generated by rotating the region bounded by $y=\sqrt{x}$ and $y=\frac{x}{2}$ around each of the axes.

Understanding the Problem

To find the volume of the solid of revolution, we need to understand the problem clearly. We are given two curves, $y=\sqrt{x}$ and $y=\frac{x}{2}$ , which bound a region. We need to rotate this region around each of the axes to form a solid and then find its volume.

Setting Up the Problem

To set up the problem, we need to identify the axis of rotation and the region to be rotated. In this case, we have two axes of rotation, the x-axis and the y-axis. We need to find the volume of the solid formed by rotating the region bounded by $y=\sqrt{x}$ and $y=\frac{x}{2}$ around each of these axes.

Finding the Volume of the Solid of Revolution

To find the volume of the solid of revolution, we need to use the method of disks or washers. This method involves dividing the region into thin disks or washers and then summing up the volumes of these disks or washers.

Volume of the Solid of Revolution around the x-axis

To find the volume of the solid of revolution around the x-axis, we need to use the method of disks. We can set up the following definite integral:

V = \pi \int_{0}^{4} \left(\sqrt{x}\right)^2 - \left(\frac{x}{2}\right)^2 \, dx

where $V$ is the volume of the solid of revolution.

Volume of the Solid of Revolution around the y-axis

To find the volume of the solid of revolution around the y-axis, we need to use the method of washers. We can set up the following definite integral:

V = 2\pi \int_{0}^{2} x \left(\sqrt{x} - \frac{x}{2}\right) \, dy

where $V$ is the volume of the solid of revolution.

Evaluating the Definite Integrals

To evaluate the definite integrals, we need to use the following formulas:

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

\int a^x \, dx = \frac{a^x}{\ln(a)} + C

where $n$ is a constant and $a$ is a constant.

Using these formulas, we can evaluate the definite integrals as follows:

Volume of the Solid of Revolution around the x-axis

V = \pi \int_{0}^{4} \left(\sqrt{x}\right)^2 - \left(\frac{x}{2}\right)^2 \, dx

V = \pi \left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}} - \frac{x^2}{4}\right]_{0}^{4}

V = \pi \left[\frac{4^{\frac{3}{2}}}{\frac{3}{2}} - \frac{4^2}{4}\right] - \pi \left[\frac{0^{\frac{3}{2}}}{\frac{3}{2}} - \frac{0^2}{4}\right]

V = \pi \left[\frac{8}{\frac{3}{2}} - 4\right]

V = \pi \left[\frac{16}{3} - 4\right]

V = \pi \left[\frac{16}{3} - \frac{12}{3}\right]

V = \pi \left[\frac{4}{3}\right]

V = \frac{4\pi}{3}

Volume of the Solid of Revolution around the y-axis

V = 2\pi \int_{0}^{2} x \left(\sqrt{x} - \frac{x}{2}\right) \, dy

V = 2\pi \int_{0}^{2} x \sqrt{x} - \frac{x^2}{2} \, dy

V = 2\pi \left[\frac{x^{\frac{5}{2}}}{\frac{5}{2}} - \frac{x^3}{6}\right]_{0}^{2}

V = 2\pi \left[\frac{2^{\frac{5}{2}}}{\frac{5}{2}} - \frac{2^3}{6}\right] - 2\pi \left[\frac{0^{\frac{5}{2}}}{\frac{5}{2}} - \frac{0^3}{6}\right]

V = 2\pi \left[\frac{4\sqrt{2}}{\frac{5}{2}} - \frac{8}{6}\right]

V = 2\pi \left[\frac{8\sqrt{2}}{5} - \frac{4}{3}\right]

V = 2\pi \left[\frac{24\sqrt{2}}{15} - \frac{20}{15}\right]

V = 2\pi \left[\frac{24\sqrt{2} - 20}{15}\right]

V = \frac{2\pi(24\sqrt{2} - 20)}{15}

Conclusion

Introduction

In our previous article, we discussed how to set up definite integrals with respect to $y$ to find the volume of the solid of revolution generated by rotating the region bounded by $y=\sqrt{x}$ and $y=\frac{x}{2}$ around each of the axes. We also evaluated the definite integrals to find the volume of the solid of revolution around each of the axes. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the volume of the solid of revolution around the x-axis?

A: The volume of the solid of revolution around the x-axis is $\frac{4\pi}{3}$ .

Q: What is the volume of the solid of revolution around the y-axis?

A: The volume of the solid of revolution around the y-axis is $\frac{2\pi(24\sqrt{2} - 20)}{15}$ .

Q: How do I set up the definite integral to find the volume of the solid of revolution around the x-axis?

A: To set up the definite integral, you need to use the method of disks. The formula for the volume of the solid of revolution around the x-axis is:

V = \pi \int_{0}^{4} \left(\sqrt{x}\right)^2 - \left(\frac{x}{2}\right)^2 \, dx

Q: How do I set up the definite integral to find the volume of the solid of revolution around the y-axis?

A: To set up the definite integral, you need to use the method of washers. The formula for the volume of the solid of revolution around the y-axis is:

V = 2\pi \int_{0}^{2} x \left(\sqrt{x} - \frac{x}{2}\right) \, dy

Q: What is the method of disks?

A: The method of disks is a technique used to find the volume of a solid of revolution. It involves dividing the region into thin disks and then summing up the volumes of these disks.

Q: What is the method of washers?

A: The method of washers is a technique used to find the volume of a solid of revolution. It involves dividing the region into thin washers and then summing up the volumes of these washers.

Q: How do I evaluate the definite integral to find the volume of the solid of revolution?

A: To evaluate the definite integral, you need to use the following formulas:

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

\int a^x \, dx = \frac{a^x}{\ln(a)} + C

where $n$ is a constant and $a$ is a constant.

Q: What is the formula for the volume of the solid of revolution around the x-axis?

A: The formula for the volume of the solid of revolution around the x-axis is:

V = \pi \int_{0}^{4} \left(\sqrt{x}\right)^2 - \left(\frac{x}{2}\right)^2 \, dx

Q: What is the formula for the volume of the solid of revolution around the y-axis?

A: The formula for the volume of the solid of revolution around the y-axis is:

V = 2\pi \int_{0}^{2} x \left(\sqrt{x} - \frac{x}{2}\right) \, dy

Conclusion

In this article, we answered some frequently asked questions related to the topic of volume of solid of revolution. We hope that this article has been helpful in clarifying any doubts you may have had. If you have any further questions, please feel free to ask.

Additional Resources

Volume of Solid of Revolution: A Comprehensive Approach
Volume of Solid of Revolution: A Comprehensive Approach - Q&A
Mathematics