Calc 2: Volume of a Torus Homework - Solve Step-by-Step

In summary, the conversation discusses solving a problem involving finding the volume of a torus by using integration. The first problem was successfully solved by taking the integral of a function. The second problem posed a challenge as it required understanding the concept of rotating a slice around a specific point. After some guidance, it was determined that the radius of rotation for a general point x can be represented by 3-x. This was then applied to the integral to successfully solve the problem.
  • #1
KevinL
37
0

Homework Statement


http://faculty.tcu.edu/richardson/Calc2/H20090323torusVolume.htm This is a link to the homework.

2. The attempt at a solution
I did number 1 by doing 2 * integral from -2 to 2 of sqrt(4-x^2) = 4pi

The second problem is where I am completely confused and don't know how to do it or even conceptualize it. Trying to look at an example of a similar problem on another site, I tried to do 4pi * integral from -2 to 2 of xsqrt(4-x^2) but it equals 0.

I think if I understand this one ill be able to do the rest of the homework.
 
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  • #2
You are close. But the x in the second integral is supposed to be the radius of rotation of the slice located at x and rotated around x=3. How far is x from the center of rotation at x=3?
 
  • #3
So it should be 4pi * integral from -2 to 2 of 3sqrt(4-x^2) ?
 
  • #4
No. If x is -2 the radius of rotation is 5. If x is +2 the radius of rotation is 1. Right? The distance of those points from x=3. What is the radius for a general point x between -2 and 2?
 
  • #5
r= 3-x
 
  • #6
You've got it. Put that into your integral instead of x.
 

FAQ: Calc 2: Volume of a Torus Homework - Solve Step-by-Step

What is a torus?

A torus is a 3-dimensional shape that looks like a donut or an inner tube. It is created by rotating a circle around an axis that is outside of the circle.

How do you calculate the volume of a torus?

The formula for calculating the volume of a torus is V = 2π²Rr², where R is the distance from the center of the torus to the center of the circular cross-section, and r is the radius of the circular cross-section.

What is the difference between a torus and a cylinder?

A torus is a 3-dimensional shape with a hole in the middle, while a cylinder is a solid shape with two circular bases and a curved side. The volume formula for a torus is also different from that of a cylinder.

Why is calculating the volume of a torus important?

Understanding the volume of a torus is important in many fields, such as engineering, architecture, and physics. It can help in designing structures and understanding the properties of rotating objects.

How do I solve the volume of a torus homework step-by-step?

To solve the volume of a torus homework, follow these steps:
1. Identify the values of R and r in the given problem.
2. Substitute the values into the formula V = 2π²Rr².
3. Simplify the equation and solve for the volume.
4. Round the answer to the appropriate number of significant figures.

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