Calculating Volume of Rotation w/ Shell Method: Integral Difference

In summary: This difference might lead to different results when attempting to integrate different functions with \pi.
  • #1
alingy1
325
0
calculate the volume obtained when the area bounded by y=e^x, x=1, y=1, is rotated around x=4. use shell method. Can I see your integral?

My teacher gave me the first integral. I found the second one here: BUT: the two do not give the same numerical result:
integral of pi((4-lnx)^2-9) FROM 1 to e =/= integral of 2pi(4-x)(e^x-1) dx from 0 to 1
 
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  • #2
alingy1 said:
calculate the volume obtained when the area bounded by y=e^x, x=1, y=1, is rotated around x=4. use shell method. Can I see your integral?

My teacher gave me the first integral. I found the second one here: BUT: the two do not give the same numerical result:
integral of pi((4-lnx)^2-9) FROM 1 to e =/= integral of 2pi(4-x)(e^x-1) dx from 0 to 1

This integral -- "integral of pi((4-lnx)^2-9) FROM 1 to e" -- is using disks (washers, actually), but the term you show as lnx should be ln(y). In a more nicely formated form, this integral is
$$ \pi \int_{y = 1}^e [(4 - ln(y))^2 - 9]dy$$
Since both integrals represent the volume of the same geometric object, they have to give the same result. Since you're not getting the same result, you must be doing something wrong. Please show us your work for the integral above.
 
  • #3
Both integrals evaluate to roughly 14.91. Where do you get a different value?
 
  • #4
  • #5
That mean you shouldn't use Wolfram for this problem! At least not until you are sure you can use it properly. As you say, Wolfram is interpreting [itex]\pi(x)[/itex] as a function "pi of x", which returns the number of primes less than or equal to x (or the largest integer less than or equal to x if x is not itself an integer) not as the number [itex]\pi[/itex] times x.
 

What is the Shell Method for calculating volume of rotation?

The Shell Method is a mathematical technique used to find the volume of a solid obtained by rotating a two-dimensional shape around a specific axis. It involves breaking the solid into thin cylindrical shells and calculating their volumes using integration.

How do I set up the integral for the Shell Method?

The integral for the Shell Method is set up as ∫2π(r)(h)dx, where r is the distance from the axis of rotation to the shell, h is the height of the shell, and dx is the thickness of the shell. This integral is then evaluated from the lower bound to the upper bound of the shape's domain.

What is the difference between using the Shell Method and the Disk Method?

The main difference between the Shell Method and the Disk Method is the shape that is being rotated. The Shell Method is used for shapes that have a hole or cavity in the center, while the Disk Method is used for shapes that are solid or do not have a hole.

Can the Shell Method be used for any shape?

No, the Shell Method can only be used for shapes that have a vertical axis of rotation and can be broken down into cylindrical shells. If the shape does not meet these criteria, a different method, such as the Washer Method, must be used to calculate the volume of rotation.

What are some common mistakes to avoid when using the Shell Method?

Some common mistakes when using the Shell Method include not setting up the integral correctly, using the wrong axis of rotation, and not properly accounting for the thickness of the shells. It is important to carefully consider the shape and axis of rotation before setting up the integral to ensure accurate results.

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