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

• blumfeld0
In summary, the problem is to find the volume of a solid resulting from a ball of radius 10 with a hole of radius 5 drilled through its center. The volume of the removed cylinder is pi*5^2*10*sqrt(3). The volume of the original sphere is 4/3*pi*10^3. The integral of pi*(100-y^2) dy from 10 to 10*sqrt(3) should be multiplied by 2 to find the volume of the remaining solid. Be sure to carefully set the limits of integration.
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!

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

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!

What is the definition of volume of solid region?

Volume of solid region refers to the three-dimensional space occupied by a solid object. It is measured in units such as cubic meters or cubic centimeters.

How is the volume of a regular solid calculated?

The volume of a regular solid is calculated by multiplying the length, width, and height of the object together. For example, the volume of a cube can be found by multiplying the length of one side by itself three times.

What about the volume of an irregular solid?

The volume of an irregular solid can be calculated using the displacement method. This involves submerging the object in a liquid and measuring the amount of liquid it displaces. The displaced liquid is then measured, and the volume of the solid can be determined.

How does the volume of a solid region relate to its mass?

The volume of a solid region is directly proportional to its mass. This means that as the volume of a solid increases, so does its mass. This relationship can be expressed in the formula density = mass/volume, where density is a constant for a given material.

Are there any real-life applications of calculating the volume of solid regions?

Yes, calculating the volume of solid regions has many practical applications in fields such as architecture, engineering, and manufacturing. It is also essential in determining the amount of materials needed for construction projects and in measuring the capacity of containers and vessels.

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