# Volume of a cone using spherical coordinates with integration

• mahrap
Then, using spherical coordinates with the given values for theta and phi, you can find the value of p. In summary, to find the volume of a cone with radius R and height H using spherical coordinates, use the equation x = p cos theta sin phi, y= p sin theta sin phi, and z= p cos phi, with theta between 0 and 2 pie and phi between 0 and pie / 4, and find the value of p by considering the coordinates of a point at the widest part of the cone.
mahrap
Find the volume of a cone with radius R and height H using spherical coordinates.

so x^2 + y^2 = z^2
x = p cos theta sin phi
y= p sin theta sin phi
z= p cos phi

I found theta to be between 0 and 2 pie
and phi to be between 0 and pie / 4.
i don't know how to find p though. how would i do this.

mahrap said:
so x^2 + y^2 = z^2
That does not fit the given dimensions.
x = p cos theta sin phi
y= p sin theta sin phi
z= p cos phi

I found theta to be between 0 and 2 pie
and phi to be between 0 and pie / 4.
i don't know how to find p though. how would i do this.
After correcting the equation for the cone, consider the coordinates of a point at the circumference at the widest part of the cone.

## 1. What is the formula for finding the volume of a cone using spherical coordinates with integration?

The formula for finding the volume of a cone using spherical coordinates with integration is V = 1/3 * π * r^2 * h, where r is the radius of the base and h is the height of the cone.

## 2. How does using spherical coordinates differ from using Cartesian coordinates to find the volume of a cone?

Using spherical coordinates allows for a more simplified integration process as the cone's curved surface can be represented by a single variable, while in Cartesian coordinates, multiple variables are required to represent the curved surface.

## 3. What are the limits of integration when using spherical coordinates to find the volume of a cone?

The limits of integration for the radius are 0 to the radius of the base of the cone, and the limits for the height are 0 to the height of the cone.

## 4. Can the formula for finding the volume of a cone using spherical coordinates be used for any type of cone?

Yes, the formula can be used for any type of cone as long as the cone's curved surface can be represented by a single variable.

## 5. Are there any real-world applications for finding the volume of a cone using spherical coordinates with integration?

Yes, this method can be used in fields such as physics, engineering, and architecture to calculate the volume of objects with curved surfaces, such as tanks, silos, and domes.

• Calculus and Beyond Homework Help
Replies
3
Views
855
• Calculus and Beyond Homework Help
Replies
7
Views
811
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
2K
• Calculus and Beyond Homework Help
Replies
6
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
9
Views
482
• Calculus and Beyond Homework Help
Replies
33
Views
3K
• Calculus and Beyond Homework Help
Replies
4
Views
503