Sphere-Cylinder Volume: Using Cylindrical Shells Method for Solid Calculation

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In summary, a cylindrical hole is drilled through the center of a sphere, and the volume of the remaining solid is found using the shell method. If you know the length of the cylindrical hole, you can determine the volume remaining without knowing any other dimensions.
  • #1
Justabeginner
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Homework Statement


A cylindrical hole of radius R is drilled through the center of a sphere of radius 3. Use the method of cylindrical shells to find the volume of the remaining solid, given that the solid is 6 cm. high.


Homework Equations


V= ∫2* ∏rh dr (shell method)


The Attempt at a Solution



I think this is the form to be used for this problem, but I am probably mistaken.
2 * ∫2* ∏ * x (R^2 - x^2) dx with a= r and b= R
4∏ ∫x (R^2 - x^2) dx

Then I'm stumped on how to do this :/ Thank you for your help.
 
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  • #2
Justabeginner said:

Homework Statement


A cylindrical hole of radius R is drilled through the center of a sphere of radius 3. Use the method of cylindrical shells to find the volume of the remaining solid, given that the solid is 6 cm. high.


Homework Equations


V= ∫2* ∏rh dr (shell method)


The Attempt at a Solution



I think this is the form to be used for this problem, but I am probably mistaken.
2 * ∫2* ∏ * x (R^2 - x^2) dx with a= r and b= R
4∏ ∫x (R^2 - x^2) dx

Then I'm stumped on how to do this :/ Thank you for your help.

You have a = r but there is no r in the problem. Do you mean a = 3? Otherwise your integral looks OK, so just integrate it.

[Edit] That should be ##\sqrt{R^2-x^2}## in that integrand.
 
Last edited:
  • #3
Oh wow, I see what I did wrong. So my final answer turns out to be

4/3 * pi (R^2 - 9) ^(3/2)

Is this correct? Thank you.
 
  • #4
Justabeginner said:
Oh wow, I see what I did wrong. So my final answer turns out to be

4/3 * pi (R^2 - 9) ^(3/2)

Is this correct? Thank you.

I didn't notice you left off the square root over the ##R^2-x^2## in your integral formula, but apparently you had it correct on your paper since that looks like the right answer.
 
  • #5
Great, thank you so much! I feel more confident about how to do these problems now :)
 
  • #6
One interesting corollary of this problem is that if you know the length of the cylindrical hole you can determine the volume remaining without knowing any other dimensions.
 
  • #7
Thank you for your insight on that. I will keep that in mind if I ever come across such a problem again.
 

1. What is the formula for calculating the volume of a sphere?

The formula for calculating the volume of a sphere is (4/3)πr^3, where r is the radius of the sphere.

2. How do you find the volume of a cylinder?

The formula for calculating the volume of a cylinder is πr^2h, where r is the radius of the base and h is the height of the cylinder.

3. Can you use the same formula to find the volume of a sphere and a cylinder?

No, the formula for calculating the volume of a sphere and a cylinder are different. The formula for a sphere is based on its radius, while the formula for a cylinder is based on its radius and height.

4. How is the volume of a sphere different from a cylinder?

The volume of a sphere is a measurement of the amount of space inside a sphere, while the volume of a cylinder is a measurement of the amount of space inside a cylinder. The formulas for calculating the volume of a sphere and a cylinder are also different.

5. Can you use the sphere-cylinder volume formula for any size of sphere or cylinder?

Yes, the sphere-cylinder volume formula can be used for any size of sphere or cylinder as long as you have the correct measurements for the radius, height, and/or diameter of the shape.

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