How Do You Calculate the Volume of a Cone Using Integration?

In summary, to find the volume of a right circular cone using integration, you must first divide the cone into horizontal slices and find the area of each slice as a function of y. Then, use the equation of the right side of the triangle to find a relationship between the radius and height of each slice. Finally, integrate with respect to y to find the total volume of the cone.
  • #1
the_storm
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Homework Statement



Using integration, Find the Volume of a right circular cone with height h and base radius r

The Attempt at a Solution


since the volume is
V(x) = [tex]\int A(x) d(x)[/tex]
so I divided the cone into horizontal circles with radius r and r = [tex]\sqrt{s^{2} + y^{2}}[/tex] where is the hypotenuse and y is the height of the cone.
then I integrate with respect to y, but I got nothing so is there any help to find the volume of the cone ?
 
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  • #2
the_storm said:

Homework Statement



Using integration, Find the Volume of a right circular cone with height h and base radius r

The Attempt at a Solution


since the volume is
V(x) = [tex]\int A(x) d(x)[/tex]
Your integral won't look like this since you are using horizontal slices, each of width dy. The area of each slice is a function of y, not x.
the_storm said:
so I divided the cone into horizontal circles with radius r and r = [tex]\sqrt{s^{2} + y^{2}}[/tex] where is the hypotenuse and y is the height of the cone.
then I integrate with respect to y, but I got nothing so is there any help to find the volume of the cone ?
Draw a vertical cross-section sketch of the cone, with the base on the horizontal axis and the vertex of the cone at (0, h). The cross section will be a triangle.

What is the equation of the right side of the triangle? You need to find a relationship between the radius of a slice and the height of the slice.
 

Related to How Do You Calculate the Volume of a Cone Using Integration?

1. 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 = (π * r^2 * h) / 3, where r is the radius of the base and h is the height of the cone.

2. How do you measure the radius and height of a right circular cone?

The radius of a right circular cone can be measured by finding the distance from the center of the base to the edge of the base. The height of a right circular cone can be measured by finding the distance from the tip of the cone to the center of the base.

3. Can the volume of a right circular cone be negative?

No, the volume of a right circular cone cannot be negative. Volume is a measure of space and cannot have a negative value.

4. How does the volume of a right circular cone change with the radius and height?

The volume of a right circular cone is directly proportional to the square of the radius and the height. This means that as the radius or height increases, the volume will also increase. Similarly, as the radius or height decreases, the volume will decrease.

5. Can you use the same formula to calculate the volume of any cone?

No, the formula for calculating the volume of a right circular cone can only be used for right circular cones. Other types of cones, such as oblique cones, have different formulas for calculating their volume.

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