How do I calculate the volume of a spherical slice using integration?

In summary, finding the volume of a spherical slice involves taking two parallel slices along lines of latitude and finding the integral of the area of each cross section. The formula for the volume of a wedge with angle θ between the two arcs is \frac{2}{3}R^3\theta.
  • #1
tandoorichicken
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How do I find the volume of any spherical slice?
 
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  • #2
What exactly do you mean by a spherical slice? My guess would be to take two half great-circles (from pole to pole) and "cut" to the line through center and both poles- although I would call that a "wedge".

If that's what you mean, then the volume depends only on the angle between the two arcs. The volume of the entire sphere is [tex]\frac{4}{3}\pi R^3[/tex]. The volume of a wedge with angle θ between the two arcs is that times [tex]\frac{\theta}{2\pi}= \frac{2}{3}R^3\theta[/tex].
 
  • #3
a slice:
say you have a circle. then you cut straight through the circle once, and then make another parallel slice. That's what I mean by a spherical slice. Basically you end up with a frying pan like solid.
 
  • #4
Ah. Instead of slicing through two lines of longitude, you slice along two lines of latitude. (Apparently we slice our apples differently!)

You will have to integrate to get that. Assume the sphere is centerd at (0,0,0) and has radius R. Take the two slices to be at z= z0 and z= z1. For each value of z between those, a cross section will be a circle centered at (0,0,z). The radius of that circle is r= √(R2- z2) and so the circle has area π(R2- z2). Taking the thickness of a thin slice to be dz, the integral becomes
π integral from z0 to z1 of (R2- z2) dz. Hmm, that's easier than I thought it would be.
 

FAQ: How do I calculate the volume of a spherical slice using integration?

What is the formula for calculating the volume of a spherical slice?

The formula for calculating the volume of a spherical slice is (1/6)πh(3r² + h²), where h is the height of the slice and r is the radius of the sphere.

How do you find the height of a spherical slice?

To find the height of a spherical slice, you can use the Pythagorean theorem to solve for h in the formula (1/6)πh(3r² + h²). Alternatively, you can use the formula h = r - √(r² - x²), where x is the distance from the center of the sphere to the plane that cuts the slice.

Can the volume of a spherical slice be negative?

No, the volume of a spherical slice cannot be negative. The formula for calculating the volume always results in a positive value, as it involves squaring and multiplying positive numbers.

What is the unit of measurement for the volume of a spherical slice?

The unit of measurement for the volume of a spherical slice is typically cubic units, such as cubic inches, cubic centimeters, or cubic meters.

Is the volume of a spherical slice the same as the volume of a cylinder with the same height and radius?

No, the volume of a spherical slice is not the same as the volume of a cylinder with the same height and radius. The formula for calculating the volume of a cylinder is πr²h, which is different from the formula for calculating the volume of a spherical slice. Additionally, the shape of a spherical slice is curved, while a cylinder is a straight, flat shape.

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