# Question about volume calculation using disks or shells

• pamparana
In summary: So you don't need to consider the negative limit. In summary, the method of shells and disks are used for calculating volumes in 3D by sweeping infinitesimal regions and disks. In the example of calculating the volume of a couldron, the parabola was modeled as the body and the path swept by the disk was modeled as a shell. The limits for the definite integral only need to be from 0 to \sqrt{a} because the dV element of the shell takes care of both sides of the parabola.
pamparana
Hello everyone,

I was going through the Single Variable Calculus lectures on the MIT Opencourseware site and looking at calculating volumes in 3D using the method of shells and disks.

In both these methods, we sweep the infinitesmal region in 3D sweeping and sweep a disk. My question is actually about setting up the limits for the definite integral to calculate the final volume.

In the example, they were looking at calculating the volume of a couldron where the body was modeled like a parabola and path sweeped by the disk was modeled as a shell. Now the parabola was symmetric around the origin and when taking the limit, they only considered one side of the parabola (so, the lower limit was 0 and the upper limit was $$\sqrt{a}$$.

My question is that should't the limit be from $$\sqrt{-a}$$ to $$\sqrt{+a}$$. There was brief discussion about this but I could not follow the reasoning which was along the lines that the sweeping dx region took care of the other half...

Would be vert grateful if someone can clarify this for me.

And a merry xmas and new year to all!

Cheers,

Luca

pamparana said:
Hello everyone,

I was going through the Single Variable Calculus lectures on the MIT Opencourseware site and looking at calculating volumes in 3D using the method of shells and disks.

In both these methods, we sweep the infinitesmal region in 3D sweeping and sweep a disk.
They don't both sweep a disk. A shell is not the same as a disk.
My question is actually about setting up the limits for the definite integral to calculate the final volume.

In the example, they were looking at calculating the volume of a couldron where the body was modeled like a parabola and path sweeped by the disk was modeled as a shell. Now the parabola was symmetric around the origin and when taking the limit, they only considered one side of the parabola (so, the lower limit was 0 and the upper limit was $$dV=\sqrt{a}$$

My question is that should't the limit be from $$\sqrt{-a}$$ to $$\sqrt{+a}$$.

You mean $-\sqrt{a}$ not $\sqrt{-a}$, but no, you don't want the negative limit. The dV element of the shell is generally

$$( 2\pi x) (y_u-y_l)dx$$

for revolution about the y axis. The $( 2\pi x)$ is the circumference of the shell at position x and the y-upper - y-lower is the height of the shell. You get both sides of the x-axis because the circumference goes all the way around.

## 1. How do you calculate the volume using disks or shells?

The volume of a solid of revolution can be calculated using the method of disks or shells. For disks, the formula is V = π∫(R(x))^2 dx, where R(x) is the radius of the disk at a given value of x. For shells, the formula is V = 2π∫xf(x)dx, where f(x) is the circumference of the shell at a given value of x.

## 2. When should you use the method of disks versus the method of shells?

The method of disks is typically used when the cross-sections of the solid are perpendicular to the axis of rotation, while the method of shells is used when the cross-sections are parallel to the axis of rotation.

## 3. Can the method of disks or shells be used for any shape?

Yes, the method of disks or shells can be used for any shape that can be rotated around an axis. It is commonly used for shapes such as cylinders, cones, and spheres.

## 4. Are there any limitations to using the method of disks or shells?

The method of disks or shells may not be accurate for solids with irregular shapes or for solids with varying cross-sections that cannot be easily integrated. In these cases, other methods such as the method of washers may be more appropriate.

## 5. How does the choice of method affect the accuracy of the volume calculation?

The choice of method may affect the accuracy of the volume calculation, as the method of disks or shells may not provide an exact solution for some complex shapes. However, using smaller and more numerous disks or shells can improve the accuracy of the calculation.

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