Finding the volume with Shell method

In summary, when using the shell method to find volume of revolution about the y-axis, the formula is V = ∫(x0 to x1)xy dx, where x is the radius of the shell and y is the length/height of the shell. If negative y-values are used, simply change the sign of the answer. When dealing with two curves, the integral should be set up as V = ∫(0 to 1)xy1 dx + ∫(1 to 2)xy2 dx, where y1 and y2 are the respective curves. The 2π term should also be included in the final formula. The disk/washer method is another way to find the volume of revolution and uses
  • #1
denian
641
0
the volume that i get is a negative value, there must be something wrong with my working.
is shell height = y + 2 - y^2
or i am wrong?
tq.
 

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  • #2
The shell Method:
When taking a volume of revolution about the y-axis, the formula is,

[tex]V = \int_{x_0} ^{x_1} xy \ dx[/tex]

where x is the radius of the shell, dx represents the shell thickness and y is the length, or height, of the shell.

Since, in this case, you are taking negative y-values, then you will get a negative answer for the volume. Simply change the sign.

You have two curves to contend with, so you should make up the integral like this,

[tex]V = \int_0 ^1 xy_1 \ dx + \int_1 ^2 xy_2 \ dx[/tex]

[tex]\mbox{where}\ y_1\ \mbox{is}\ \sqrt{x}\ \mbox{and}\ y_2\ \mbox{is}\ (x-2).[/tex]
 
  • #3
thank you, but I am a bit confused.
im doing some self-study here, and the formula they wrote in the book is what i wrote in the first line.
btw, the x-axis is the axis of rotation.

nvm. i try to figure it out again :)
 
  • #4
Fermat said:
The shell Method:
When taking a volume of revolution about the y-axis, the formula is,

[tex]V = \int_{x_0} ^{x_1} xy \ dx[/tex]

where x is the radius of the shell, dx represents the shell thickness and y is the length, or height, of the shell.

Since, in this case, you are taking negative y-values, then you will get a negative answer for the volume. Simply change the sign.

You have two curves to contend with, so you should make up the integral like this,

[tex]V = \int_0 ^1 xy_1 \ dx + \int_1 ^2 xy_2 \ dx[/tex]

[tex]\mbox{where}\ y_1\ \mbox{is}\ \sqrt{x}\ \mbox{and}\ y_2\ \mbox{is}\ (x-2).[/tex]

...don't you mean:

[tex]V=\mathbf{2\pi}\int_{a}^{b}xf(x)dx[/tex]
 
  • #5
apmcavoy said:
...don't you mean:

[tex]V=\mathbf{2\pi}\int_{a}^{b}xf(x)dx[/tex]
yes,

thanks for the correction,

y = f(x)

and I completely missed out the 2pi !
 
  • #6
denian said:
thank you, but I am a bit confused.
im doing some self-study here, and the formula they wrote in the book is what i wrote in the first line.
btw, the x-axis is the axis of rotation.

nvm. i try to figure it out again :)
Sorry about the confusion there. I'm afraid I misread your work. Actually, I'd never heard of the shell method before.

You working is correct. The shell height/length is (y + 2 - y²). And you will have gotten a volume of -5pi/6 units, yes?
I got a volume of +5pi/6 units using my volume of revolution method - which I've now found out is called the disk method or washer method.

You are getting a negative value simply because you are using negative y-values for the radius, but +ve values for the shell height.
 

1. What is the shell method?

The shell method is a technique used to find the volume of a solid of revolution, where the cross-sections are formed by shells (hollow cylindrical shapes) instead of discs or washers.

2. When is the shell method used?

The shell method is typically used when rotating a function around a vertical axis, or when the function is difficult to integrate using other methods such as the disc or washer method.

3. How is the volume calculated using the shell method?

The volume is calculated by integrating the circumference of the shell, multiplied by the height of the shell, over the desired interval. This can be represented by the formula V = 2π∫(radius)(height)dx.

4. What are the steps for using the shell method?

The steps for using the shell method are:

  1. Identify the axis of rotation and the desired interval.
  2. Set up the integral using the formula V = 2π∫(radius)(height)dx.
  3. Express the radius and height of the shell in terms of x.
  4. Integrate the expression, using proper limits of integration.
  5. Simplify the integral and evaluate for the final volume.

5. What are some common mistakes when using the shell method?

Some common mistakes when using the shell method include:

  • Forgetting to square the radius in the formula.
  • Using the wrong limits of integration.
  • Forgetting to multiply by 2π in the formula.
  • Not expressing the radius and height in terms of x.
  • Not simplifying the integral correctly.

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