Integrating Mass of a Hollow Sphere: Multivariable Calculus Explained

In summary, the formula for calculating the mass of a hollow sphere is M = ρ * V, where ρ represents the density of the material and V represents the volume of the sphere. To integrate the mass of a hollow sphere using multivariable calculus, you need to set up a triple integral with the appropriate bounds for the radius, height, and angle of the sphere. This integral will be evaluated using the formula M = ρ * V. The main difference between a solid sphere and a hollow sphere in terms of mass integration is that the latter requires a multivariable integral due to varying densities. Multivariable calculus is necessary for integrating the mass of a hollow sphere because it allows for the calculation of the mass of a three
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cwill53
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Homework Statement
I'm just trying to prove that gravitational force on an object within a hollow sphere is zero when these are the only two objects under consideration within the system.
Relevant Equations
$$F_g=G\frac{mM}{r^2}$$
I know some multivariable calculus, I just want someone to walk me through the integration deriving the mass element dM and the integration of thin rings composing the hollow sphere. It would also be nice if you could show me doing it one way using the solid angle and one way without using the solid angle. Thanks.
 
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1. What is the formula for calculating the mass of a hollow sphere?

The formula for calculating the mass of a hollow sphere is M = ρV, where ρ is the density and V is the volume.

2. How do you find the volume of a hollow sphere?

The volume of a hollow sphere can be found using the formula V = (4/3)π(r3 - R3), where r is the radius of the outer sphere and R is the radius of the inner sphere.

3. What is multivariable calculus?

Multivariable calculus is a branch of mathematics that deals with the study of functions of several variables. It involves the use of derivatives and integrals to analyze and solve problems in multiple dimensions.

4. How is the mass of a hollow sphere integrated in multivariable calculus?

The mass of a hollow sphere is integrated by setting up a triple integral with the density function as the integrand and the volume of the hollow sphere as the limits of integration. This involves integrating with respect to three variables (x, y, and z) to account for the three-dimensional nature of the problem.

5. What are some real-world applications of integrating the mass of a hollow sphere?

Integrating the mass of a hollow sphere has many practical applications, such as calculating the weight of a hollow metal ball used in sports, determining the density of a planet or other celestial body, and analyzing the structural integrity of a hollow sphere-shaped object in engineering and architecture.

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