Find the volume by using shell and disk method

In summary, the shell and disk method is a technique used in calculus to find the volume of a solid of revolution. It involves breaking down the solid into infinitesimally thin disks or cylindrical shells and integrating their volumes. This method can be used for any solid of revolution with a defined axis of rotation, and is particularly useful for complex shapes. The main difference between the shell and disk method is the shape of the elements used for calculation. The equation for finding the volume is V = ∫2πrh dx, and it can be modified depending on the orientation of the solid and axis of rotation. This method can also be used for solids with varying cross-sectional areas, but may require more complex integration and slicing of elements.
  • #1
jaychay
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Can you check it for me that I done it right or not ?
Thank you in advance.
 
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  • #2
shells ...

$\displaystyle V = 2\pi \int_0^1 (x+2) \cdot x \, dx + 2\pi \int_1^4 (x+2) \cdot \sqrt{x} \, dx$

washers ...

$\displaystyle V = \pi \int_0^1 6^2 - (y+2)^2 \, dy + \pi \int_1^2 6^2 - (y^2+2)^2 \,dy$
 

Related to Find the volume by using shell and disk method

What is the shell method for finding volume?

The shell method is a mathematical technique for finding the volume of a solid of revolution. It involves slicing the solid into thin cylindrical shells and adding up the volumes of all the shells to get the total volume.

What is the disk method for finding volume?

The disk method is another mathematical technique for finding the volume of a solid of revolution. It involves slicing the solid into thin disks and adding up the volumes of all the disks to get the total volume.

When should I use the shell method versus the disk method?

The choice between using the shell method or the disk method depends on the shape of the solid of revolution. If the solid has a hole or a gap in the middle, the shell method is more appropriate. If the solid is a solid cylinder or has a solid center, the disk method is more suitable.

What are the steps for using the shell method to find volume?

The steps for using the shell method to find volume are as follows:1. Identify the axis of revolution.2. Determine the limits of integration.3. Set up the integral using the formula for the volume of a cylindrical shell.4. Evaluate the integral to find the total volume.

What are the steps for using the disk method to find volume?

The steps for using the disk method to find volume are as follows:1. Identify the axis of revolution.2. Determine the limits of integration.3. Set up the integral using the formula for the volume of a circular disk.4. Evaluate the integral to find the total volume.

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