Problem 44 - Essential Calculus

Question

Volume of a Torus } Repeat Exercise 43 for a torus formed by revolving the region bounded by the circle \(x^{2}+y^{2}=r^{2}\) about the line \(x=R,\) where \(r

Step-by-Step Solution

Answer

The volume of the torus is given by the formula \(V=2 \times \pi^2 \times r^2 \times (R-r)\)

Step 1: Define Variables

Define r as radius of the circle and R as the line about where the region is revolving to form the torus. Assume that r

Step 2: Apply Formula for Volume of Torus

Use the formula to find the volume of a torus: \(V=2 \times \pi^2 \times r^2 \times (R-r)\). This represents the volume of a donut-shaped object.

Step 3: Simplify Expression

Here we do not have specific values for R and r, so the expression can't be calculated further. The final volume of the torus formed by revolving the region bounded by the circle \(x^2 + y^2 = r^2\) around the line \(x = R\) is given by \(V=2 \times \pi^2 \times r^2 \times (R-r)\)

Key Concepts

Circular Region

A circular region is the space enclosed by a circle. In our problem, this circle is defined by the equation \( x^{2} + y^{2} = r^{2} \). This equation represents a circle centered at the origin \((0,0)\) with a radius \( r \). The circle's radius is the constant \( r \), which measures the distance from any point on the circle to its center. This circular region is key in forming the shape of the torus when it is revolved around a line. Understanding this enclosed space helps visualize how it will eventually contribute to the torus's overall volume.

Revolution about a line

Revolution about a line refers to the process of rotating a shape, like a circle, around a straight line to create a 3-dimensional object. In this exercise, the circle revolves around the vertical line \( x = R \). This line is outside the original circle, with \( R > r \), ensuring the circle does not intersect the axis of rotation.
Imagine tracing the path of the circle as it revolves completely around the line. This movement creates a torus, or a doughnut shape. The center of this doughnut is at \( x = R \), and it is symmetric around this vertical line. Understanding revolution around a line allows us to determine the shape's dimensions and, subsequently, its volume.

Torus formula

The torus formula is a mathematical expression used to calculate the volume of a torus formed by revolving a circular region.
The formula is given by \( V = 2 \times \pi^2 \times r^2 \times (R-r) \). Let's break it down:

• \(r^2\) represents the area of the circular region being revolved.
• \(R-r\) is the distance from the centricity of the circle to the line of revolution, which determines the 'largeness' of the torus.
• \(2\times\pi^2\) is a constant that accounts for the rotation creating a full torus.

This formula is essential as it ties the geometric attributes of the circle and the path it takes during revolution into one useful measurement: volume.

Volume Calculation

Volume calculation involves using the torus formula with given values. Since the problem did not provide specific numbers for \( R \) and \( r \), the volume remains expressed as \( V = 2 \times \pi^2 \times r^2 \times (R-r) \). This expression represents the potential volume for any circle of radius \( r \) revolving around a line at distance \( R \).
This general formula allows for flexibility. We can plug in different numerical values for \( R \) and \( r \) to find the volume of various tori. Knowing how to use this formula is crucial in applications ranging from engineering to design where exact volumetric calculations are necessary.

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Problem 45

Other exercises in this chapter

Problem 44

The level of sound \(\beta\) (in decibels) with an intensity of \(I\) is $$\beta(I)=10 \log _{10} \frac{I}{I_{0}}$$ where \(I_{0}\) is an intensity of \(10^{-16

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Problem 44

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Problem 45

In Exercises \(45-48,\) (a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) explain why the area of the region is difficu

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Problem 45

The value of a tract of timber is\(V(t)=100,000 e^{0.8 \sqrt{t}}\) where \(t\) is the time in years, with \(t=0\) corresponding to 1998 . If money earns interes

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Key Concepts

Circular Region

A circular region is the space enclosed by a circle. In our problem, this circle is defined by the equation \( x^{2} + y^{2} = r^{2} \). This equation represents a circle centered at the origin \((0,0)\) with a radius \( r \). The circle's radius is the constant \( r \), which measures the distance from any point on the circle to its center. This circular region is key in forming the shape of the torus when it is revolved around a line. Understanding this enclosed space helps visualize how it will eventually contribute to the torus's overall volume.

Revolution about a line

Revolution about a line refers to the process of rotating a shape, like a circle, around a straight line to create a 3-dimensional object. In this exercise, the circle revolves around the vertical line \( x = R \). This line is outside the original circle, with \( R > r \), ensuring the circle does not intersect the axis of rotation.
Imagine tracing the path of the circle as it revolves completely around the line. This movement creates a torus, or a doughnut shape. The center of this doughnut is at \( x = R \), and it is symmetric around this vertical line. Understanding revolution around a line allows us to determine the shape's dimensions and, subsequently, its volume.

Torus formula

The torus formula is a mathematical expression used to calculate the volume of a torus formed by revolving a circular region.
The formula is given by \( V = 2 \times \pi^2 \times r^2 \times (R-r) \). Let's break it down:

• \(r^2\) represents the area of the circular region being revolved.
• \(R-r\) is the distance from the centricity of the circle to the line of revolution, which determines the 'largeness' of the torus.
• \(2\times\pi^2\) is a constant that accounts for the rotation creating a full torus.

This formula is essential as it ties the geometric attributes of the circle and the path it takes during revolution into one useful measurement: volume.

Volume Calculation

Volume calculation involves using the torus formula with given values. Since the problem did not provide specific numbers for \( R \) and \( r \), the volume remains expressed as \( V = 2 \times \pi^2 \times r^2 \times (R-r) \). This expression represents the potential volume for any circle of radius \( r \) revolving around a line at distance \( R \).
This general formula allows for flexibility. We can plug in different numerical values for \( R \) and \( r \) to find the volume of various tori. Knowing how to use this formula is crucial in applications ranging from engineering to design where exact volumetric calculations are necessary.