Volume of Solids of Revolution

In summary, the problem asks to find the volume of the solid formed by rotating the region between y=cosx and y=0 on the interval [0,pi] around the line x=1. There is confusion about whether to rotate around x=1 or y=1, but ultimately it is determined that the rotation around x=1 would include the right half of the curve and cannot be ignored. The solution is complex and unconventional.
  • #1
Econometricia
33
0
Find the volume of the solid st,
1. y=cos x , y= 0 in [0,pi] ; Rotated around x=1




2. I am slightly confused, I see that the area will double around twice so I can just use the left half of the curve. I am just not sure how to do so.
 
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  • #2
Are you sure that you copied this problem correctly. The line x = 1 does not divide the region between y = cosx and y = 0 into two equal halves, so rotating it around the line x = 1 makes a complicated volume of revolution. Could it be that you're supposed to rotate the region around the line y = 1?
 
  • #3
Yes, not 2 equal halves but the rotation around the x=1 axis would cover the right hand side of the curve in rotation and double over so can't we ignore that piece?
 
  • #4
No, I don't think so. The part of the region under y = cosx on [0, 1] would cover the part on [1, pi/2], so I suppose you could ignore that part. The part of the curve on [pi/2, pi] is below the x-axis. The part above [0, 1] is above the x-axis. This is a very unusual problem.
 

What is the "Volume of Solids of Revolution"?

The volume of solids of revolution is a mathematical concept used to calculate the volume of 3D shapes created by rotating a 2D shape around an axis. This is often used in calculus to find the volume of irregular shapes.

How do you calculate the volume of a solid of revolution?

To calculate the volume of a solid of revolution, you first need to determine the region of the 2D shape that will be rotated around the axis. Then, use the formula V = π∫(f(x))^2 dx, where f(x) represents the function of the curve and dx is the infinitesimal width of the slices that make up the solid.

What are some real-life applications of calculating the volume of solids of revolution?

Calculating the volume of solids of revolution has various real-life applications, such as in engineering, architecture, and manufacturing. It is used to determine the amount of material needed to create certain shapes, such as pipes, bottles, and car parts.

Are there any limitations to using the volume of solids of revolution?

While the volume of solids of revolution is a useful mathematical concept, there are limitations to its application. It can only be used for shapes that can be rotated around an axis, and it assumes the shape is continuous and has a smooth curve.

How does the volume of solids of revolution relate to other mathematical concepts?

The volume of solids of revolution is closely related to other mathematical concepts, such as integrals, derivatives, and the shell method. It is also used in conjunction with the Pythagorean theorem and the formula for the volume of a cylinder.

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