# Disk, Washer, Shell Multiple Integrals

• whatlifeforme
In summary, for the given problem of determining how many integrals are required for disk, washer, and shell methods, it appears that two integrals will be needed for the disk and washer method, while only one integral will be needed for the shell method. This is based on the fact that the region to be rotated has no breaks or abnormalities, and different shapes can be seen when using disks/washers versus cylindrical shells. Drawing sketches of the region and solid can also help in determining the number of integrals needed.
whatlifeforme

## Homework Statement

Determine how many integrals are required for disk, washer, and shell method.

## Homework Equations

x=3y^2 - 2 and x=y^2 from (-2,0) to (1,1) about x-axis.

## The Attempt at a Solution

Since there are no breaks or abnormalities in the graph it appears that 1 integral will solve for each to me. I have no other way of looking at this problem.

how do i know when the disk method requires more than one integral?

Last edited:
whatlifeforme said:

## Homework Statement

Determine how many integrals are required for disk, washer, and shell method.

## Homework Equations

x=3y^2 - 2 and x=y^2 from (-2,0) to (1,1)

## The Attempt at a Solution

Since there are no breaks or abnormalities in the graph it appears that 1 integral will solve for each to me. I have no other way of looking at this problem.

how do i know when the disk method requires more than one integral?
What axis is this to be revolved around ?

SammyS said:
What axis is this to be revolved around ?

evidently it will take two for disk and washer, and one for shell, but I'm not sure why.

whatlifeforme said:

evidently it will take two for disk and washer, and one for shell, but I'm not sure why.
Have you drawn a sketch of the region that will be rotated?
Have you drawn a different sketch of the solid that results from the rotation?

If you haven't drawn both sketches, it will be very difficult to work this problem.

If you use disks/washers of width Δx, your 2nd drawing should show that they change shape at (0, 0), because the upper and lower edges change. This means you will need two integrals.

If you use cylindrical shells of width Δy, the left and right edges don't change, so a single integral will suffice.

## 1. What is the difference between disk, washer, and shell multiple integrals?

These are all methods for calculating the volume of a solid using multiple integrals. The main difference is the shape of the cross-section used to slice the solid. Disk integrals use circular cross-sections, washer integrals use annular (ring-shaped) cross-sections, and shell integrals use cylindrical cross-sections.

## 2. When should I use disk, washer, or shell multiple integrals?

Disk and washer integrals are typically used for solids of revolution, where the shape of the solid can be obtained by rotating a curve around an axis. Shell integrals are used for solids with a cylindrical or prismatic shape. The choice of method depends on the shape of the solid and the cross-sections used to slice it.

## 3. How do I set up the integral for disk, washer, or shell multiple integrals?

The general formula for these integrals is ∫ab π(R(x))2 dx, where R(x) is the radius of the cross-section at a given x-value. The limits of integration, a and b, will depend on the shape and orientation of the solid.

## 4. What are some common mistakes to avoid when solving disk, washer, or shell multiple integrals?

One common mistake is not properly setting up the integral by choosing the correct limits of integration and function to integrate. It is also important to correctly identify the shape and orientation of the solid to determine which method to use. Additionally, make sure to double check the units of measurement to ensure the final answer is in the correct units.

## 5. How are disk, washer, and shell multiple integrals related to each other?

Disk and washer integrals can be thought of as special cases of shell integrals, where the height of the cylindrical cross-section is infinitesimally small. This means that disk and washer integrals can be obtained by integrating a shell integral using the appropriate limits and function. Additionally, the volume calculated using different methods should be the same for the same solid.

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