Integral to find the volume of a torus

In summary, to find the volume of a torus formed by rotating a circle around the y-axis, you can use Pappus' theorem by multiplying the distance traveled by the center of the circle (2πr) by the area of the circle (πr^2). In this case, the volume is 16π^2.
  • #1
togame
18
0

Homework Statement


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.


Homework Equations


Formula for the semi-circle: [itex]y=\sqrt{4-x^2}[/itex]
Solving for x give [itex]x=\pm\sqrt{4-y^2}[/itex]


The Attempt at a Solution


I was thinking of solving the integral from -2 to 2 using both the positive and negative sides, then evaluate the integral and then multiply by 2, but I'm not so sure about my equation.

[tex]2\int_{-2}^2\pi\big( (3-\sqrt{4-y^2})^2 - (3+\sqrt{4-y^2})^2 \big) \mathrm d y[/tex]

I just seem lost on this one :(
 
Physics news on Phys.org
  • #2
togame said:

Homework Statement


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.

Homework Equations


Formula for the semi-circle: [itex]y=\sqrt{4-x^2}[/itex]
Solving for x give [itex]x=\pm\sqrt{4-y^2}[/itex]

The Attempt at a Solution


I was thinking of solving the integral from -2 to 2 using both the positive and negative sides, then evaluate the integral and then multiply by 2, but I'm not so sure about my equation.
[tex]2\int_{-2}^2\pi\big( (3-\sqrt{4-y^2})^2 - (3+\sqrt{4-y^2})^2 \big) \mathrm d y[/tex]
I just seem lost on this one :(
I believe the equation is:

[tex]V=\int_{-2}^2\pi\big( (3+\sqrt{4-y^2})^2 - (3-\sqrt{4-y^2})^2 \big) \mathrm d y[/tex]

[tex]V=\int_{-2}^2\pi\cdot 12(\sqrt{4-y^2}) \mathrm d y[/tex]

then trigonometric substitution...
 
  • #3
togame said:

Homework Statement


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.

Homework Equations


Formula for the semi-circle: [itex]y=\sqrt{4-x^2}[/itex]
Solving for x give [itex]x=\pm\sqrt{4-y^2}[/itex]

The Attempt at a Solution


I was thinking of solving the integral from -2 to 2 using both the positive and negative sides, then evaluate the integral and then multiply by 2, but I'm not so sure about my equation.

[tex]2\int_{-2}^2\pi\big( (3-\sqrt{4-y^2})^2 - (3+\sqrt{4-y^2})^2 \big) \mathrm d y[/tex]

I just seem lost on this one :(

Edit : Got beat to it.
 
  • #4
the way to do this is to use pappus' theorem. the product of the distance traveled by the center of the circle by the area of the circle.
 

1. What is a torus?

A torus is a geometric shape that resembles a donut or a ring. It is a three-dimensional object with a circular cross-section and a hole in the center.

2. How is the volume of a torus calculated using integral?

The volume of a torus can be calculated by using the integral of a function that represents the cross-sectional area of the torus. This integral can be evaluated using the formula V = 2π²Rr², where R is the radius of the torus and r is the radius of the circular cross-section.

3. What is the significance of using integral to find the volume of a torus?

Using integral to find the volume of a torus allows for a more accurate estimation of the volume, as it takes into account the varying cross-sectional areas of the torus. It also allows for the calculation of volumes for more complex torus shapes.

4. Can the volume of a torus be calculated without using integral?

Yes, the volume of a torus can also be calculated using other methods such as the formula V = (π²Rr)(2R-r), or by approximating the torus as a series of smaller cylinders. However, using integral provides a more precise and generalizable solution.

5. Are there any real-world applications of using integral to find the volume of a torus?

Yes, integral is commonly used in engineering and physics to calculate the volume of objects with complex shapes, such as torus-shaped pipes or containers. It is also used in computer graphics to generate 3D models of torus-shaped objects.

Similar threads

  • Calculus and Beyond Homework Help
Replies
10
Views
289
  • Calculus and Beyond Homework Help
Replies
12
Views
3K
  • Calculus and Beyond Homework Help
Replies
14
Views
572
  • Calculus and Beyond Homework Help
Replies
20
Views
385
  • Calculus and Beyond Homework Help
Replies
34
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
936
  • Calculus and Beyond Homework Help
Replies
12
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
459
  • Calculus and Beyond Homework Help
Replies
27
Views
2K
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
Back
Top