Let { S $}$ Be The Region Enclosed By The Graphs Of { Y = 3x $}$ For { 0 \leq X \leq 1 $}$.8. What Is The Volume Of The Solid Obtained By Rotating Region { S $}$ About The X-axis?Given
Introduction
In mathematics, the volume of a solid obtained by rotating a region about the x-axis can be calculated using the method of disks or washers. This method involves integrating the area of the circular cross-sections of the solid with respect to the x-axis. In this article, we will explore how to calculate the volume of a solid obtained by rotating a region enclosed by the graphs of y = 3x for 0 ≤ x ≤ 1 about the x-axis.
The Method of Disks
The method of disks is a technique used to calculate the volume of a solid obtained by rotating a region about the x-axis. This method involves integrating the area of the circular cross-sections of the solid with respect to the x-axis. The formula for the volume of a solid obtained by rotating a region about the x-axis using the method of disks is:
V = π∫[a,b] (f(x))^2 dx
where V is the volume of the solid, π is a mathematical constant approximately equal to 3.14, f(x) is the function that defines the region, and a and b are the limits of integration.
Calculating the Volume of the Solid
In this problem, we are given the region enclosed by the graphs of y = 3x for 0 ≤ x ≤ 1. To calculate the volume of the solid obtained by rotating this region about the x-axis, we can use the method of disks. The function that defines the region is f(x) = 3x, and the limits of integration are a = 0 and b = 1.
Using the formula for the volume of a solid obtained by rotating a region about the x-axis using the method of disks, we can calculate the volume of the solid as follows:
V = π∫[0,1] (3x)^2 dx
To evaluate this integral, we can use the power rule of integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C. Applying this rule to the integral, we get:
V = π∫[0,1] 9x^2 dx
V = π(9/3)x^3 | [0,1]
V = π(3)x^3 | [0,1]
V = π(3)(1^3 - 0^3)
V = π(3)
V = 3Ï€
Therefore, the volume of the solid obtained by rotating the region enclosed by the graphs of y = 3x for 0 ≤ x ≤ 1 about the x-axis is 3π cubic units.
Conclusion
In this article, we have explored how to calculate the volume of a solid obtained by rotating a region about the x-axis using the method of disks. We have applied this method to a specific problem involving the region enclosed by the graphs of y = 3x for 0 ≤ x ≤ 1 and have calculated the volume of the solid obtained by rotating this region about the x-axis. The result is 3π cubic units.
References
- [1] "Calculus: Early Transcendentals" by James Stewart
- [2] "Calculus" by Michael Spivak
Further Reading
For further reading on the topic of calculating the volume of a solid obtained by rotating a region about the x-axis, we recommend the following resources:
- [1] "The Method of Disks" by Wolfram MathWorld
- [2] "Volume of a Solid" by Math Open Reference
Glossary
- Method of Disks: A technique used to calculate the volume of a solid obtained by rotating a region about the x-axis.
- Volume of a Solid: The amount of three-dimensional space enclosed by a solid.
- X-Axis: The horizontal axis in a coordinate system.
- Region: A set of points in a coordinate system that satisfy certain conditions.
Q&A: Volume of a Solid Obtained by Rotating a Region About the X-Axis ====================================================================
Introduction
In our previous article, we explored how to calculate the volume of a solid obtained by rotating a region about the x-axis using the method of disks. In this article, we will answer some frequently asked questions about this 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 the x-axis. This method involves integrating the area of the circular cross-sections of the solid with respect to the x-axis.
Q: How do I calculate the volume of a solid using the method of disks?
A: To calculate the volume of a solid using the method of disks, you need to follow these steps:
- Define the region that you want to rotate about the x-axis.
- Find the function that defines the region.
- Determine the limits of integration.
- Use the formula for the volume of a solid obtained by rotating a region about the x-axis using the method of disks: V = π∫[a,b] (f(x))^2 dx
- Evaluate the integral to find the volume of the solid.
Q: What is the formula for the volume of a solid obtained by rotating a region about the x-axis using the method of disks?
A: The formula for the volume of a solid obtained by rotating a region about the x-axis using the method of disks is:
V = π∫[a,b] (f(x))^2 dx
where V is the volume of the solid, π is a mathematical constant approximately equal to 3.14, f(x) is the function that defines the region, and a and b are the limits of integration.
Q: How do I evaluate the integral in the formula for the volume of a solid obtained by rotating a region about the x-axis using the method of disks?
A: To evaluate the integral in the formula for the volume of a solid obtained by rotating a region about the x-axis using the method of disks, you can use the power rule of integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C. You can also use other integration techniques, such as substitution or integration by parts.
Q: What are some common mistakes to avoid when calculating the volume of a solid using the method of disks?
A: Some common mistakes to avoid when calculating the volume of a solid using the method of disks include:
- Not defining the region correctly
- Not finding the function that defines the region
- Not determining the limits of integration correctly
- Not evaluating the integral correctly
- Not using the correct formula for the volume of a solid obtained by rotating a region about the x-axis using the method of disks
Q: Can I use the method of disks to calculate the volume of a solid obtained by rotating a region about the y-axis?
A: No, the method of disks is used to calculate the volume of a solid obtained by rotating a region about the x-axis. To calculate the volume of a solid obtained by rotating a region about the y-axis, you need to use the method of washers.
Q: What is the method of washers?
A: The method of washers is a technique used to calculate the volume of a solid obtained by rotating a region about the y-axis. This method involves integrating the area of the annular cross-sections of the solid with respect to the y-axis.
Conclusion
In this article, we have answered some frequently asked questions about the volume of a solid obtained by rotating a region about the x-axis using the method of disks. We hope that this article has been helpful in clarifying any confusion about this topic.
References
- [1] "Calculus: Early Transcendentals" by James Stewart
- [2] "Calculus" by Michael Spivak
Further Reading
For further reading on the topic of calculating the volume of a solid obtained by rotating a region about the x-axis, we recommend the following resources:
- [1] "The Method of Disks" by Wolfram MathWorld
- [2] "Volume of a Solid" by Math Open Reference
Glossary
- Method of Disks: A technique used to calculate the volume of a solid obtained by rotating a region about the x-axis.
- Volume of a Solid: The amount of three-dimensional space enclosed by a solid.
- X-Axis: The horizontal axis in a coordinate system.
- Region: A set of points in a coordinate system that satisfy certain conditions.
- Method of Washers: A technique used to calculate the volume of a solid obtained by rotating a region about the y-axis.