Volume inside a sphere and cone

In summary, to find the volume inside x^2 + y^2 + z^2 =2z and inside z^2 = x^2 + y^2, the problem can be divided into two parts: the volume above the z axis and the volume below the z axis. By completing the square, the first part can be represented as a sphere of radius one shifted above the z axis by one. By converting to spherical coordinates, the limits for the integral can be found and the integral can be solved. The second part involves a cone and the volume can be calculated by subtracting the cone volume from the half sphere volume.
  • #1
timnswede
101
0
Find the volume laying inside x^2 + y^2 + z^2 =2z and inside z^2 = x^2 + y^2.

This is a problem my professor made, so I have no way of checking my answer.
What I did first was completed the square for the sphere and got x^2 + y^2 + (z-1)^2 = 1, which is a sphere of radius one shifted above the z axis by one. and z^2 = x^2 + y^2 is two cones, but I am only worried about the one above the z axis.
I changed x^2 + y^2 + (z-1)^2 = 1 into spherical coordinates: (ρ^2)sin^2(Φ) + (ρ^2)cos^2(Φ) - 2ρcosΦ = 0 and got ρ = 2cosΦ.
So the limits for θ are from 0 to 2pi, ρ is from 0 to 2cosΦ, and since I am going from the top of the sphere to the bottom of the cone above the z axis I think my limits for Φ are from 0 to pi/2.
So my integral is ∫(0,2pi)∫(0,pi/2)∫(0,2cosΦ) (ρ^2)sinΦ dρdΦdθ which is a pretty simple integral. Does this look right? I'm not 100% sure on my ρ and Φ limits.
 
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  • #2
You can see that for z=1 both equations become x2+y2=1 and for z>0 sphere lays below the "cone". Two surfaces makes two volumes the sum of them is the total sphere volume. Changing z by z-1 both equations the sphere have the circle x2+y2=1 on the x-y plane. Now is easy to separate integral limits with azimuth but you need not this.
Calculate the cone volume (must be negative), convert to positive and add or remove from the half sphere volume.
 
Last edited:
  • #3
If I make a drawing, I see your cone and sphere. I also see they intersect at z = 1. Wouldn't it be easier to separate z > 1 and z < 1 ?

[edit] whoa! Theo was a little faster !
 

Related to Volume inside a sphere and cone

What is the formula for finding the volume of a sphere?

The formula for finding the volume of a sphere is V = (4/3)πr³, where V is the volume and r is the radius of the sphere.

How does the volume of a sphere compare to the volume of a cone?

The volume of a cone is exactly one-third the volume of a sphere with the same radius. This means that if you were to fill three cones with water and pour it into a sphere, it would completely fill the sphere.

What are the units of measurement for volume?

The units of measurement for volume can vary depending on the system of measurement being used. In the metric system, the standard unit is cubic meters (m³), while in the imperial system, the standard unit is cubic inches (in³).

Can the volume of a sphere or cone be negative?

No, the volume of a sphere or cone cannot be negative. Volume is a measure of the amount of space an object occupies, and it cannot be less than zero.

How is the volume of a cone calculated?

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

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