How Do You Calculate the Volume Between a Cone and a Sphere?

In summary, the problem asks to find the volume of the solid that lies above the mantle surface of a cone with an angle of π/3 and below a sphere with a radius of 4cosΦ. This can be solved using a triple integral with the bounds 0≤ρ≤4cosΦ, 0≤Φ≤π, and 0≤θ≤2π, where ρ represents the distance from the origin, Φ represents the angle from the z-axis, and θ represents the angle from the projection in the xy plane. However, different textbooks may define the θ and Φ coordinates differently.

Homework Statement

Find volume of the solid that lies above the cone Φ = π/3 and below the sphere ρ = 4cosΦ

The Attempt at a Solution

Obviously this is a triple integral. My book tells me that 0 ≤ρ≤ 4cosΦ

but this makes no sense to me.

From the problem, it lies ABOVE the cone Φ = π/3 and below the sphere ρ = 4cosΦ, so wouldn't that imply that ρ is not starting at 0?

What I did was solved ρ = 4cosΦ

arccos(ρ/4) = Φ and set it = to pi/3

arccos(ρ/4) = π/3

ρ/4 = cosπ/3

ρ = 4 * (1/2) = 2

so wouldn't 2≤ρ≤4cosΦ be the bounds? I don't understand how the lower bound can start at 0 when its asking for what's above the cone and below the sphere..

wouldnt that imply that ρ is not starting at 0?

I suggest you draw an image of what things look like.

Orodruin said:

I suggest you draw an image of what things look like.

ρ is the distance to the point from the origin
Φ is the angle from the z axis to the point. 0≤Φ≤pi
θ is the angle to the point from the projection in the xy plane.

I did draw an image, it looks like an icecream cone basically. The part I want is the icecream on the top of the cone. I still don't understand why ρ is 0..

Hmm I think I may be graphing it wrong. Maybe I have no clue how to graph ρ = 4cosΦ..

ρ is the distance to the point from the origin
Φ is the angle from the z axis to the point. 0≤Φ≤pi
θ is the angle to the point from the projection in the xy plane.
Different textbooks treat spherical coordinates in different ways, particularly the ##\theta## and ##\phi## coordinates. According to this wikipedia article (https://en.wikipedia.org/wiki/Spherical_coordinate_system), physics books consider ##\phi## to be the angle in the "x-y" plane, while math textbooks consider ##\phi## to be measured from the positive z-axis.

The part I want is the icecream on the top of the cone.
No, this is not correct. The "cone" that the problem talks about is the mantle surface of the cone. If it was just the top that was intended, the problem would talk about a plane, not about a cone.

What is the definition of volume?

Volume is the amount of space occupied by an object or substance, typically measured in cubic units such as cubic meters or cubic centimeters.

How do you calculate the volume of a solid?

The formula for finding the volume of a solid depends on the shape of the object. For a rectangular solid, the volume is calculated by multiplying the length, width, and height. For a cylinder, the volume is calculated by multiplying the area of the base by the height. For irregularly shaped solids, the volume can be found by using the water displacement method or by using a formula specific to that shape.

What is the difference between volume and mass?

Volume and mass are two different measurements. Volume is a measure of the amount of space occupied by a substance, while mass is a measure of the amount of matter in a substance. In other words, volume is the physical size of an object, while mass is the amount of material it contains.

Why is it important to find the volume of a solid?

Knowing the volume of a solid is important in many scientific and real-world applications. It can help determine the amount of material needed for construction or manufacturing, the capacity of containers, the dosage of medication, and the density of a substance. Additionally, volume is a key factor in many mathematical and scientific calculations.

What are some common units used to measure volume?

The most commonly used units to measure volume are cubic meters (m3), cubic centimeters (cm3), liters (L), and milliliters (mL). In chemistry, volume is often measured in cubic decimeters (dm3), or liters, while in the United States, gallons (gal) and fluid ounces (fl oz) are commonly used to measure volume.

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