Volumes by Cylindrical Shells (Calculus II)

In summary, the question asks for the volume of a solid obtained by rotating the region between y = x and y = x^2 about the y-axis. The integral set up is correct, with the length of a shell represented by 2pi x, the height by x-x^2, and the thickness by dx.
  • #1
shamieh
539
0
Quick question, may seem rather dumb - but I just want to make sure of something..

Question: Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x^2

so when I am setting up my integral am I correct in saying \(\displaystyle TOP - BOTTOM i.e. --> \int^1_0 (2\pi x) (x - x^2) dx\)?
 
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  • #2
Re: Volumes by Cyllindrical Shells (Calculus II)

It looks good to me. The $2\pi x$ gives you the length of a shell, were you to straighten it out. The $x-x^{2}$ gives you the height of a shell, and the $dx$ gives you the thickness.
 

FAQ: Volumes by Cylindrical Shells (Calculus II)

What is the formula for finding the volume of a shape using cylindrical shells?

The formula for finding the volume of a shape using cylindrical shells is V = 2π ∫(R(x) - r(x)) dx, where R(x) is the outer radius, r(x) is the inner radius, and the integral is taken over the region of interest.

How is the formula for volumes by cylindrical shells derived?

The formula for volumes by cylindrical shells is derived by taking the cross-section of a solid shape and revolving it around an axis to create cylindrical shells. The volume of these shells can then be calculated using basic geometry and the sum of these volumes gives the total volume of the shape.

How is the concept of cylindrical shells useful in calculus?

The concept of cylindrical shells is useful in calculus as it allows for the calculation of volumes of irregular shapes that are difficult to find using traditional methods. It also helps in solving problems involving volumes of revolution, where the shape is formed by rotating a curve around an axis.

Can the formula for volumes by cylindrical shells be applied to 3-dimensional shapes?

Yes, the formula for volumes by cylindrical shells can be applied to 3-dimensional shapes, as long as the shape can be approximated by many thin cylindrical shells. This method is particularly useful for finding the volume of shapes with curved or irregular boundaries.

Are there any limitations to using the formula for volumes by cylindrical shells?

One limitation of using the formula for volumes by cylindrical shells is that it only works for shapes with rotational symmetry. Additionally, the shape must be able to be approximated by many thin cylindrical shells for the formula to accurately calculate the volume.

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