# Find the volume using shell and disk method

• MHB
• jaychay
In summary, the shell method and disk method are two techniques used to find the volume of a solid of revolution. The main difference between them is the shape of the infinitesimal slices used. The method used depends on the shape of the solid and the axis of revolution. The formula for finding the volume using the shell method is V = 2π ∫a^b x (f(x) - g(x)) dx. Both methods can be used for any type of solid of revolution, but there may be limitations for solids with irregular shapes.
jaychay
I have tried to do it but I end up getting the wrong answer.

$\displaystyle V = 18\pi - 2\pi \int_0^{\pi/2} x \cos{x} \, dx$

$\displaystyle V = 18\pi - \pi \int_0^1 [\arccos{y}]^2 \, dy$

Last edited by a moderator:
skeeter said:
$\displaystyle V = 18\pi - 2\pi \int_0^{\pi/2} x \cos{x} \, dx$

$\displaystyle V = 18\pi - \pi \int_0^1 [\arccos{y}]^2 \, dy$
Can you tell me where did 18 pi come from ?

jaychay said:
Can you tell me where did 18 pi come from ?

volume of a hemisphere is $\dfrac{2\pi r^3}{3}$ and $r = 3$

## What is the difference between the shell and disk method for finding volume?

The shell method involves using the radius of a cylindrical shell to find the volume of a solid of revolution, while the disk method involves using the radius of a disk to find the volume. The shell method is typically used for solids with a hole in the middle, while the disk method is used for solids without a hole.

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

As mentioned before, the shell method is best used for solids with a hole in the middle, while the disk method is used for solids without a hole. Additionally, the shape of the solid of revolution can also determine which method is more appropriate to use. It is recommended to visualize the solid and determine which method would be more efficient in finding the volume.

## What is the formula for finding volume using the shell method?

The formula for finding volume using the shell method is V = 2π∫(radius)(height)(thickness)dr, where the radius is the distance from the axis of revolution to the shell, the height is the height of the shell, and the thickness is the difference between the outer and inner radius of the shell.

## What is the formula for finding volume using the disk method?

The formula for finding volume using the disk method is V = π∫(radius)^2dx, where the radius is the distance from the axis of revolution to the disk and dx is the width of the disk. This formula can be used for both solids with a hole and without a hole.

## Can the shell and disk method be used for any shape?

The shell and disk method can be used for any shape that can be formed by rotating a function around an axis. This includes shapes such as circles, parabolas, and hyperbolas. However, the methods may be more complex for certain shapes and may require additional calculations.

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