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Chapter 6: Problem 71

A torus (doughnut) Find the volume of the torus formed when the circle of radius 2 centered at (3,0) is revolved about the y-axis. Use geometry to evaluate the integral. A torus (doughnut) Find the volume of the torus formed when the circle of radius 2 centered at (3,0) is revolved about the y-axis. Use geometry to evaluate the integral.

Short Answer

Expert verified
Answer: The volume of the torus is 24 * pi^2 cubic units.

Step by step solution

01

Identify the circle's equation and radius

The equation of a circle with radius 2 centered at (3,0) can be found using the equation (x-a)^2 + (y-b)^2 = r^2. In our case, a = 3, b = 0, and r = 2. So, the equation of the circle is: (x-3)^2 + y^2 = 4
02

Identify the variables for the torus volume

The volume formula for a torus is V = 2 * pi * R * pi * r^2, where R is the distance from the center of the circle to the axis of rotation (in our case, the y-axis) and r is the radius of the circle.
03

Find the distance R

Since the circle is centered at (3,0), the distance R from the center to the y-axis is 3 units. Therefore, R = 3.
04

Find the radius r of the torus

We are given that the radius of the circle is 2. So, the radius r of the torus is 2.
05

Calculate the volume of the torus

Now that we have the values for R and r, we can calculate the volume using the formula V = 2 * pi * R * pi * r^2: V = 2 * pi * 3 * pi * 2^2 V = 6 * pi^2 * 4 V = 24 * pi^2 So, the volume of the torus is 24 * pi^2 cubic units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rotational Solids
When you think about rotational solids, imagine taking a flat shape, like a circle, and spinning it around an axis to create a 3D object. It's like twirling a baton or rolling clay to form a shape.
In geometry and calculus, these solids are often considered because they help understand how shapes behave in three dimensions, especially when predicting volume and surface area.
In this problem, we explore a torus, the scientific term for a doughnut-shaped solid.
  • The formation of a torus involves revolving a circle around a line outside the circle.
  • For a torus, this process involves rotating a circle around the y-axis, creating a donut-like shape.
Understanding rotational solids is a fundamental concept in geometry and helps bridge the understanding of flat shapes into 3D realities.
Integration Techniques
Integration techniques are powerful tools in mathematics to find areas and volumes, especially when shapes become complex, like a torus.
In our exercise, we compute the volume of a torus by integrating the circle's equation over a circular path.
This involves using the following concepts:
  • Recognize that integrating helps sum up infinitely small pieces to find a total.
  • Understand the setup of known formulas, like the torus volume formula, to streamline calculations.
For the torus volume, the geometry simplifies the complex integration into a form that's more manageable, making use of \( V = 2 \pi R \pi r^2 \). Once Rusty in ellipses, integrating can directly compute volumes or areas for more intricate landscapes.
Geometry in Calculus
Geometry in calculus melds shapes with dynamic calculations, allowing deeper insights into forms and sizes through calculus processes.
In the volume of a torus, geometry steps in to provide an intuitive balance between shapes and calculations. For instance,
  • The circle's equation \((x-3)^2 + y^2 = 4\) establishes a foundational geometric description.
  • Using these coordinates, calculus translates this into a calculable form by applying integral concepts.
By understanding these geometry-based formulas, like the radius and central distance, we leverage spatial layouts with exact computations.
These concepts harmonize to form solid foundations, whether predicting space inside a doughnut or examining planetary pathways.

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