- #1
robertjford80
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Homework Statement
I'm just interested in knowing where the 4 comes from in front of the integral.
Measuring volume of spheres using triple integrals
- Thread starter robertjford80
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In summary, to calculate the volume of a sphere using triple integrals, you need to set up the integral using spherical coordinates and integrate over the entire sphere. The formula for the volume of a sphere using triple integrals is ∫∫∫ρ^2sinφdρdφdθ, where ρ is the radius, φ is the angle from the z-axis, and θ is the angle from the x-axis. This differs from using double integrals to measure volume, which are used for flat surfaces and use rectangular coordinates. Triple integrals can be used to measure the volume of any 3-dimensional shape as long as it can be described mathematically. There are various tools available to help visualize triple integr
I'm just interested in knowing where the 4 comes from in front of the integral.
- #2
tiny-tim
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hi robertjford80! 
that 4∫0π/2 is really ∫02π …
i suppose the writer thought it looks confusingly like ∫00, though i see nothing wrong with it!
that 4∫0π/2 is really ∫02π …
i suppose the writer thought it looks confusingly like ∫00, though i see nothing wrong with it!
- #3
robertjford80
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Ok, pretty unexpected but I guess it makes sense.
- #4
HallsofIvy
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In other words, they are using the circular symmetry of the figures, integrating from 0 to [itex]\pi/2[/itex], rather than from 0 to [itex]2\pi[/itex], then multiplying by 4.
FAQ: Measuring volume of spheres using triple integrals
How do you calculate the volume of a sphere using triple integrals?
To calculate the volume of a sphere using triple integrals, you need to set up the integral using spherical coordinates and integrate over the entire sphere. The formula for the volume of a sphere using triple integrals is ∫∫∫ρ^2sinφdρdφdθ, where ρ is the radius, φ is the angle from the z-axis, and θ is the angle from the x-axis.
What is the difference between using double integrals and triple integrals to measure volume?
Double integrals are used to find the volume of a 3-dimensional shape that is bounded by a flat surface, while triple integrals are used to find the volume of a 3-dimensional shape that is bounded by curved surfaces, such as a sphere. Triple integrals also require the use of spherical or cylindrical coordinates, while double integrals use rectangular coordinates.
Can triple integrals be used to measure the volume of any 3-dimensional shape?
Yes, triple integrals can be used to measure the volume of any 3-dimensional shape as long as the shape can be described using a mathematical formula or equation. However, the setup of the integral may be more complex for shapes with curved surfaces.
Is there a way to visualize triple integrals for measuring the volume of a sphere?
Yes, there are various software programs and online tools that allow you to visualize triple integrals for measuring the volume of a sphere. These tools can help you understand the setup of the integral and how the different variables (ρ, φ, θ) affect the final volume calculation.
Are there any real-life applications of using triple integrals to measure volume?
Yes, triple integrals are commonly used in physics, engineering, and other fields to calculate the volume of objects with curved surfaces, such as spheres, cylinders, and cones. They are also used in computer graphics and animation to create 3-dimensional objects.