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

[Justabeginner]

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.

 

[LCKurtz]

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.

 

[Justabeginner]

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.

 

[LCKurtz]

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.

 

[Justabeginner]

Great, thank you so much! I feel more confident about how to do these problems now :)

 

[haruspex]

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.

 

[Justabeginner]

Thank you for your insight on that. I will keep that in mind if I ever come across such a problem again.