How do I find the volume of a described solid using integration?

In summary, the conversation was about finding the integral of a solid flipped on its side with the x-axis going through its center. The integral was from 0 to h of (pi)(r)^2, but it was suggested to relate r to h and integrate over h. The cross sectional radius, ρ(y), was also discussed and the conversation ended with the solution being found.
  • #1
Jet1045
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0

Homework Statement


I uploaded of a picture of the question so hopefully it comes up here.

Homework Equations





The Attempt at a Solution



OK! so i am SO confused on where to start.
I am imagining the solid flipped on its side with the x-axis going through its center.

So all i have is that the integral would be from 0 to h of (pi)(r)^2
Is this at all close?
Any hints would be greatly appreciated. :)
 

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  • #2
That's a start but you have to relate r to h and then integrate over h.
 
  • #3
r, R and h are given as constants in your diagram. Let's not integrate over any of them. Let y be the distance from the bottom of your solid. So y goes from 0 to h. Then your integral is the integral of (pi)(ρ(y))^2*dy for y from 0 to h. Where ρ(y) is the cross sectional radius of your solid at the height y. ρ(0)=R, ρ(h)=r. Can you figure out an expression for ρ(y) at a general height y?
 
  • #4
I GOT IT !
thanks for the help dick :)
 

FAQ: How do I find the volume of a described solid using integration?

What is the formula for finding the volume of a described solid?

The formula for finding the volume of a described solid depends on the shape of the solid. For example, the formula for a cube is V = s^3, where s is the length of one side. Other common formulas include V = lwh for a rectangular prism and V = πr^2h for a cylinder.

Can the volume of a described solid be negative?

No, the volume of a described solid cannot be negative. Volume is a measure of the space occupied by an object, so it must always be a positive value.

How accurate do my measurements need to be when calculating the volume of a described solid?

The accuracy of your measurements will depend on the precision of the tool you are using and the level of accuracy required for your specific calculation. For example, if you are measuring the volume of a small object, you may need more precise measurements than if you were measuring the volume of a large container.

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

The volume of a described solid does not directly relate to its mass. However, the density of a substance can be calculated by dividing its mass by its volume. This relationship is important in determining the mass of an object based on its volume.

Can the volume of a described solid change?

Yes, the volume of a described solid can change. For example, a balloon can increase in volume when it is inflated, and an ice cube can decrease in volume when it melts. Changes in temperature, pressure, or physical state can also cause changes in volume.

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