# 7.7 Volume Washer Methodap Calculus

### How do you find the volume of the region bounded by #y=6x# #y=x# and #y=18# is revolved about the y axis?

## 1 Answer

The “washer” method uses one integral to find the volume of the solid. I use two integrals, finding the answer as the volume of a solid minus the volume of the hole. I have found that when they set up these problems using two integrals, my students understand better what each part of the integral, especially the integrand, represents. In line with this, we will consider the formula for the Washer Method as: V = π ∫ a b (f (x)) 2 - (g (x)) 2 d x where f (x) as function of the outer radius, R g (x) as a function of the inner.

#### Explanation:

The region is the bounded region in:

graph{(y-6x)(y-x)(y-0.0001x-18) sqrt(81-(x-9)^2)sqrt(85-(y-9)^2)/sqrt(81-(x-9)^2)sqrt(85-(y-9)^2) = 0 [-28.96, 44.06, -7.7, 28.83]}

Taking vertical slices and integrating over

Rewrite the region:

As

The greater radius is

Evaluate

(Steps omitted because once it is set up, I think this is a straightforward integration.)

## Related topic

Questions

## Related questions

This method is known as Cylindrical Shells or the Shell Method.

In our previous lecture, we discussed the disk and washer method and came up with just *one formula to handle all types of cases*.

In this lesson, we will use the **Calculus Shell Method** to find the volume of a solid of revolution. But, we use this method for specific cases when we cannot use the disk and washer method.

The formula for the Shell Method comes from the idea that we are going to take a bounded region and spin or revolve infinitely thin cylinders about an axis or line.

To generate the formula, we have to remember how to find circumference, area, and volume.

## 7.7 Volume Washer Method Ap Calculus Calculator

## 7.7 Volume Washer Methodap Calculus Calculator

This method differs from the Disk & Washer method which uses rings. Now we can use any shape for the cross-section with one caveat. The shape must be able to expand or contract to cover the solid we’re evaluating, as stated in Paul’s Online Notes.

Moreover, compared to the Disk & Washer Method this method has a **significant distinction.**

For the disk/washer method, the slice is perpendicular to the axis of revolution, whereas, for the shell method, the slice is parallel to the axis of revolution.

Just look at it graphically in the detailed diagrams below:

In the video lesson below, we will walk through three classic examples. These examples provide a solid understanding of how and when to use this shell method.

## Solids of Revolution – Shell Method Video

## Solids of Revolution – Shell Method – 3 Examples

Shell Method Examples

## 7.7 Volume Washer Methodap Calculus Problems

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