Volume of Right Circular Cone: How to Calculate Using Integration

That equation is not ##x=y##. So, no, the integral is not ##\pi y^2##. In summary, the correct way to find the volume of a right circular cone with height h and base radius r is to use the method of disks, calculating the volume of a disc of thickness dy located at some y between 0 and h. The equation of the straight line representing the side of the cone must be used to determine the variable radius of the disc. The integral should be set up as πx²dy, with x² being the equation for the radius at each y value. The resulting integral should be solved and the final answer should be πh³/3.
  • #1
kent davidge
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Homework Statement



Find the volume of a right circular cone with height h and base radius r.

Homework Equations

The Attempt at a Solution



30a5on6.jpg

I've chosen the case where y = x = r = h. Then, I solve the integral in the image above and I got the book's answer. But is it actually correct?
 
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  • #2
kent davidge said:

Homework Statement



Find the volume of a right circular cone with height h and base radius r.

Homework Equations

The Attempt at a Solution



30a5on6.jpg

I've chosen the case where y = x = r = h. Then, I solve the integral in the image above and I got the book's answer. But is it actually correct?
No. y and x are not constant and not equal to h.
 
  • #3
But I've considered several cylinders along y-axis from the origin to h and each of them has radius r = x. What is the problem here?
 
  • #4
You are given ##r## is the radius of the base of the cone. You can't use it as a variable radius. And you can't assume ##y=x## making it a 45 degree cone. That isn't given.
 
  • #5
SO what can I do to find the volume?
 
  • #6
I would suggest the method of disks. Calculate the volume of a disc of thickness dy located at some y between 0 and h. You will need the equation of the straight line with the correct slope for the side to figure out the variable radius of the disc.
 
  • #7
Use this equation to work out the volume
 

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  • #8
kent davidge said:
But I've considered several cylinders along y-axis from the origin to h and each of them has radius r = x. What is the problem here?
The problem is you're being incredibly sloppy with the notation, your reasoning is horribly off, or both.

##r## and ##h## represent dimensions of the cone. So you could assume ##r=h## and get a result, but that result is only good for when ##r=h## (in which case, you don't really need two variables). You can't now claim the expression you derived is good for any right cone with arbitrary dimensions.

When you write something like "y=x=r=h", it doesn't really make sense. What do ##y## and ##x## represent here? If we take seriously that ##y=r## and ##x=r##, they're constants because ##r## is a constant (the radius of the base), so how can you integrate over either variable?
 
  • #9
LCKurtz said:
I would suggest the method of disks. Calculate the volume of a disc of thickness dy located at some y between 0 and h. You will need the equation of the straight line with the correct slope for the side to figure out the variable radius of the disc.

Faiq said:
Use this equation to work out the volume

Ok, I will do that

vela said:
The problem is you're being incredibly sloppy with the notation, your reasoning is horribly off, or both.

vela said:
You can't now claim the expression you derived is good for any right cone with arbitrary dimensions.

So a more "correct" way would be to integrate πy² over the interval [0, h], where y² = x² is the radius of each cylinder with width dy? The result would be πh³ / 3. But if I've chosen f(x) = y = x, then h would be equal to r (radius of the cone) in the end of the interval.
 
  • #10
kent davidge said:
Ok, I will do thatSo a more "correct" way would be to integrate πy² over the interval [0, h], where y² = x² is the radius of each cylinder with width dy? The result would be πh³ / 3. But if I've chosen f(x) = y = x, then h would be equal to r (radius of the cone) in the end of the interval.

As we have pointed out, ##y\ne x##. The radius of the disc is the ##x## value at that ##y##. You need the equation of the straight line representing the side to get ##x## in terms of ##y##.
 
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Related to Volume of Right Circular Cone: How to Calculate Using Integration

What is the formula for calculating the volume of a right circular cone?

The formula for calculating the volume of a right circular cone is V = (1/3)πr²h, where r is the radius of the circular base and h is the height of the cone.

What is the difference between a right circular cone and an oblique cone?

A right circular cone has a circular base that is perpendicular to the axis of the cone, while an oblique cone has a base that is not perpendicular to the axis.

How is the volume of a right circular cone related to its height?

The volume of a right circular cone is directly proportional to its height. This means that if the height is doubled, the volume will also be doubled.

What are the units of measurement for the volume of a right circular cone?

The units of measurement for the volume of a right circular cone will depend on the units used for the radius and height. For example, if the radius is measured in inches and the height is measured in feet, the volume will be measured in cubic inches.

Can the volume of a right circular cone be negative?

No, the volume of a right circular cone cannot be negative. It is always a positive value, as it represents the amount of space inside the cone.

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