Opinion? Solids of Revolution/Integration Techniques

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In summary, a student in a calculus course is looking for a problem that involves both Solids of Revolution and Integration Techniques (Parts, Substitution, or Partial Fractions). They have chosen a vase image as their problem and are unsure if it can be solved using one of the Integration Techniques. Another student suggests using the double angle formula for cos2x, but the original student is unsure if it can be applied. After further discussion, it is determined that none of the Integration Techniques can be used for this specific problem and the student decides to look for another problem, possibly involving arc lengths.
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Newbatmath
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Homework Statement



Hey everyone. In my calc course I need to create (or find on the web) a problem and apply the below two concepts that I learned in the class.

Concepts:
Solids of Revolution.
Integration Techniques: Parts, Substitution, or Partial Fractions.

Homework Equations



The problem I chose was:

http://curvebank.calstatela.edu/volrev/vase7.gif

The Attempt at a Solution



Well, the question I am asking is basically if this equation I found can be solved using one of the Integration Techniques. I was thinking Parts. Would that be right?
 
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  • #2
Just expand it out. Then use the double angle formula for cos2x, to replace sin2x
 
  • #3
So One of the Integration Techniques can't be used, then?
 
  • #4
Integration by parts might make it more complex, substitution, I don't think will work and you can't split it into partial fractions.
 
  • #5
Well, cruds.

Thank you! I'm off to find another problem then I guess. They did say I could also choose arc lengths. Maybe I'll go with that...
 

1. What are solids of revolution?

Solids of revolution are three-dimensional shapes that are formed by rotating a two-dimensional shape around a fixed axis. This process is often used in integral calculus to find the volume of irregularly shaped objects.

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

The volume of a solid of revolution can be calculated using the formula V = π∫(R(x))^2dx, where R(x) is the radius of the rotated shape at a given point. This integral can be evaluated using various integration techniques, such as the disk method, shell method, or washer method.

3. What is the difference between the disk method and shell method?

The disk method is used to find the volume of a solid of revolution when the cross-sections of the shape are perpendicular to the axis of rotation. The shell method, on the other hand, is used when the cross-sections are parallel to the axis of rotation. The shell method is often more efficient for shapes with holes or gaps.

4. Can solids of revolution be applied to real-world problems?

Yes, solids of revolution have many practical applications in fields such as engineering, physics, and architecture. For example, the volume of a water tank or the surface area of a cylindrical building can be calculated using solids of revolution.

5. What are some common integration techniques used in finding the volume of solids of revolution?

Some common integration techniques used in finding the volume of solids of revolution include the disk method, shell method, washer method, and the method of cylindrical shells. It is important to choose the appropriate technique based on the shape and orientation of the solid being rotated.

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