Can volume of a rotated function be calculated using definite integrals?

In summary, the volume of any function rotated once about the y-axis can be calculated by multiplying the definite integral of the function by 2*pi*r, where r is the size of the closed interval in the integral. This method is known as the Theorem of Pappus and can be applied to any definite integral.
  • #1
Vodkacannon
40
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Can I calculate the volume of any function rotated once about the y-axis by multiplying the definite integral of that function by 2*pi*r?

For example if we want to generate a solid 3d shape from the function -x^2+1 we multiply the integral of it, (x - x^3 / 3), by 2*pi*1. The reason r is one in this case is because the points where the function crosses the x-axis are 1 unit away from the y axis: (-1,0) and (1,0.)

(I have not taken a calculus class yet, this is just my personal reasoning.)
 
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  • #2
Check out the Theorem of Pappus.
 
  • #3
Thanks. It also turns out that r in 2*pi*r is just the size of the closed interval in the integral. So in theory this should work for any definite integral.
 

1. What is a volume integral?

A volume integral is a mathematical concept used in physics and engineering to calculate the total volume of a three-dimensional object. It involves dividing the object into infinitesimally small sections and summing up the contributions from each section to determine the total volume.

2. How is the volume integral calculated?

The volume integral is calculated using a mathematical formula known as the triple integral. This involves integrating the function representing the object's volume over the three dimensions of length, width, and height.

3. What are some applications of volume integrals?

Volume integrals have various applications in fields such as physics, engineering, and mathematics. They are used to calculate the mass, center of mass, and moment of inertia of an object, as well as to solve problems involving fluid flow and heat transfer.

4. What are the differences between a volume integral and a surface integral?

A volume integral calculates the total volume of a three-dimensional object, while a surface integral calculates the area of a two-dimensional surface. Volume integrals are used to find the volume of a solid object, whereas surface integrals are used to calculate the flux of a vector field across a surface.

5. Are there any practical considerations when using volume integrals?

When using volume integrals, it is important to consider the limitations of the mathematical model being used. For example, the object being analyzed may not have a perfectly smooth surface, or its shape may not be accurately represented by the mathematical function being integrated. These factors can affect the accuracy of the calculated volume.

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