Using Pappus' Theorem to Find Volumes of Solids of Revolution

In summary, we discussed finding the area and volume of a solid formed by rotating the region "R" under the graph of y = x^3 from x=0 to x=2 about the y-axis. We also found the first moment of area of R with respect to the y-axis and determined the x coordinate of the centroid of R. Finally, we confirmed the theorem of Pappus by showing that the volume of the solid equals the area of the region being rotated times the distance the centroid of the region travels.
  • #1
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The Region "R" under the graph of y = x^3 from x=0 to x=2 is rotated about the y-axis to form a solid.

a. Find the area of R.
b. Find the volume of the solid using vertical slices.
c. Find the first moment of area of R with respect to the y-axis. What do you notice about the integral?
d. Find the x coordinate of the centroid of R.
e. A theorem of Pappus states that the volume of a solid of revolution equals the area of the region being rotated times the distance the centroid of the region travels. Show that this problem confirms this theorem.

The Attempt at a Solution


I was able to do part "a" as the integral from 0 to 2 of x^3 dx. Also I believe part "b" is pi*[3y^(5/3)/5] evaluated from 0 to 2
 
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  • #2
I don't think "b" is right. How did you get that?
 
  • #3
Well I posted what I got. Was hoping you would show me where I went wrong...
 
  • #4
It's best if you show how you got what you got. Otherwise we just have to guess how you got what you got. That's not fun. Use the method of shells to find the volume. V=integral(2*pi*x*height*dx).
 

1. What is Pappus Theorem?

Pappus Theorem is a mathematical principle that states that the volume of a solid formed by rotating a 2-dimensional shape around an axis can be found by multiplying the area of the shape by the distance traveled by its centroid.

2. Who is Pappus and why is this theorem named after him?

Pappus of Alexandria was a Greek mathematician who lived in the 3rd century AD. He is credited with discovering this theorem, which he used to solve problems related to finding the volume of solids and the center of gravity of 2-dimensional shapes.

3. How is Pappus Theorem used in real-world applications?

Pappus Theorem has various applications in engineering and physics, such as calculating the volume of objects with irregular shapes, determining the center of gravity of structures, and analyzing the motion of rotating bodies.

4. Can Pappus Theorem be generalized to higher dimensions?

Yes, Pappus Theorem can be generalized to higher dimensions. In 3-dimensional space, it is known as Guldin's Theorem, and in n-dimensional space, it is known as the Generalized Pappus Theorem.

5. Are there any limitations to Pappus Theorem?

While Pappus Theorem is a powerful tool in solving certain types of problems, it does have limitations. It can only be applied to objects with rotational symmetry and cannot be used to find volumes of solids with holes or voids.

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