Is My Integral Calculation Correct for Finding the Volume of a Drilled Sphere?

[blumfeld0]

so i have one problem and i just need to know if my integral is right. any help would be greatly appreciated

1. a ball of radius 10 has a round hole of radius 5 drilled through its center. find the volume of the resulting solid.

i know volume of cylinder removed is pi*5^2*10*sqrt(3)

because h^2/4 + 5^2=10^2 so h = sqrt(300)= 10*sqrt(3)

and i know the volume of original sphere is 4/3*pi*10^3

and now what?
i think i need to do integral of pi*(100-y^2) dy from 10 to 10*sqrt(3) and multiply this whole answer by 2
is that right?

if so what do i do with these three pieces of information? thank you!

 

[arildno]

I advise you to be meticulous in setting up the limits of integration.

 

[tacman]

Yes, your integral is correct. You are essentially finding the volume of the solid by subtracting the volume of the cylinder (resulting from drilling the hole) from the volume of the original sphere. So, you can use the formula for the volume of a sphere (4/3*pi*r^3) and subtract the volume of the cylinder (pi*r^2*h) from it.

So, the final integral would be: V = 4/3*pi*10^3 - 2*pi*(100-y^2) dy from 10 to 10*sqrt(3).

To solve this integral, you can use the Power Rule or the Substitution Method. Once you have the final answer, you can simplify it and get the volume of the resulting solid.

Hope this helps!