Using integrals to get volume, center of mass, and surface area

In summary: How do I get the Jacobian?In summary, the problem involves finding the volume, center of mass, and surface area of a homogeneous ice-cream cone given in spherical coordinates. To find the volume, a triple integral must be used with limits in theta, phi, and rho. The center of mass can be found using a double integral of the density function. To find the surface area, a double integral of the magnitude of rrho X rphi must be used. The limits of integration can be changed to spherical coordinates using the Jacobian.
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Homework Statement




For the homogeneous ice-cream cone that is given in spherical coordinates by rho= pi/4 (the bottom part) and phi=cos(rho) (the top part), find the volume, the center of mass, and the surface area. ((You have to do this problem using integrals, known formulas from elementary geometry can be helpful, but they are not sufficient for justifying the answer)).


Homework Equations




According to spherical coordinates, "rho" is distance from the origin and "phi" is the angle between the z-axis and the line connecting the origin and the point.


The Attempt at a Solution




I know that to find the volume, I must do a triple integral. The outside integral's limits will be in theta (the angle between the line from the origin to the point and the x-axis), the middle integral's limits in terms of phi, and the inside in terms of rho. I do not know what to put inside the triple integral.

For the center of mass, I must use the double integral of the density function, and then I think I can get it from there, but I am not sure how to get the density function.

To find the surface area, I believe I must do the double integral of the magnitude of vector rrho X rphi. How do I get these vectors??

Need major help! Thanks!
 
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  • #2
To find the volume, you just integrate over 1dxdydz. Then you replace dxdydz with your spherical coordinates' Jacobian to have much nicer limits of integration
 
  • #3
How do I go about switching the limits to spherical coordinates
 

FAQ: Using integrals to get volume, center of mass, and surface area

What is the purpose of using integrals to find volume, center of mass, and surface area?

The purpose of using integrals in this context is to calculate the properties of three-dimensional objects, such as their volume, center of mass, and surface area. This is especially useful when working with irregularly shaped objects that cannot be easily measured using traditional geometric formulas.

How do you use integrals to find volume?

To find the volume of an object using integrals, you would first need to set up a triple integral that represents the volume of the object. This integral would have limits of integration corresponding to the dimensions of the object. Then, you would solve the integral to obtain the volume of the object.

How do you use integrals to find the center of mass?

The center of mass of an object can be found by setting up a triple integral that represents the mass of the object. The limits of integration would correspond to the dimensions of the object, and the integrand would be the product of the object's density and its distance from a chosen reference point. Solving the integral would give you the coordinates of the center of mass.

How do you use integrals to find surface area?

To find the surface area of an object using integrals, you would need to set up a double integral that represents the surface area of the object. The limits of integration would correspond to the dimensions of the object, and the integrand would be the square root of the sum of the squares of the partial derivatives of the object's equation with respect to each variable. Solving the integral would give you the surface area of the object.

What are some real-world applications of using integrals to find volume, center of mass, and surface area?

Integrals are used in fields such as physics, engineering, and architecture to calculate the properties of three-dimensional objects. For example, in architecture, integrals can be used to determine the amount of material needed to construct a building, while in physics, they can be used to analyze the distribution of mass in a three-dimensional object. Additionally, integrals can be used in computer graphics and animation to create realistic 3D models of objects.

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