# Finding/sketching volume with the washer method

• silicon_hobo
In summary, the washer method can be used to find the volume of a solid obtained by rotating a region about the x-axis. First, the region is bounded by two curves, y=x^2+1 & y=3-x^2. Second, the integral for volume is found by rotating the region about the x-axis. Third, a diagram is drawn to indicate the "strip" and its slice.
silicon_hobo
[SOLVED] Finding/sketching volume with the washer method

## Homework Statement

Consider the region bounded by the curves $$y=x^2+1$$ & $$y=3-x^2$$. a) Using the disk/washer method, find the volume of the solid obtained by rotating this region about the x-axis. b) Setup the integral for finding the volume of the solid obtained by rotating this region about the x-axis. In each case, draw a diagram, indicating a typical "strip" (and its slice).

## The Attempt at a Solution

I'm pretty sure the first one is right wrt x.
http://www.mcp-server.com/~lush/shillmud/inta1.6.JPG

Is this the correct way to set it up wrt y? Is there a simpler way?
http://www.mcp-server.com/~lush/shillmud/intb1.6.JPG

Also, what should I draw to "indicate a 'strip'"? I see the words strip, slab, and slice often used in a seemingly interchangeable fashion. What are they asking for here? Thanks for reading.

Hello there.

Your setup wrt has one mistake; It should be: $$\pi\int{[(3-x^2)^2-(x^2+1)^2]dx}$$

As for the second part, ideally you would not want to set this up wrt y, for the following two reasons:

1) It changes your method for finding volume from the washer method to cylindrical shells (assuming that your "slice" is perpendicular to the y-axis).

2) Given my assumption about your slice in #1, you'd have to set up 3 integrals in order to do the problem, which is pretty insane.

As for your last question, a "slice" or "strip" is simply the distance from one curve to the other for any given value of your variable of integration. For instance, take the point x=0.5. Draw a line or a narrow rectangle from the curve $$y=3-x^2$$ to the curve $$y=x^2 + 1$$ at that point. That is your slice/strip. The width of that strip is $$dx$$.

Last edited:
Thanks for the reply. I believe I have corrected my answer wrt x based on your suggestion although the new answer of 16pi seems a little bit large for the size of the rotated region. Also, I'm still not sure how to set it up wrt y. I know it's the harder way to do it but I would prefer to learn (and the question does ask for it). Cheers.

Last edited:

## 1. What is the washer method and how is it used to find/sketch volume?

The washer method is a technique used in calculus to find the volume of a solid of revolution. It involves slicing a solid into thin disks or washers and adding up the volumes of all the washers to find the total volume. This method is useful for finding the volume of irregular shapes that cannot be easily calculated using other methods.

## 2. How do you set up the washer method for finding volume?

To set up the washer method, you need to first identify the axis of rotation and the boundaries of the solid. Then, you need to choose a variable to represent the radius of the washers and an expression to represent the height of each washer. Finally, you integrate the expression with respect to the variable and multiply by π to find the total volume.

## 3. Can you give an example of using the washer method to find volume?

For example, let's say we have a solid formed by rotating the area between the curves y = x² and y = 2x about the x-axis. To find the volume using the washer method, we would set up the integral as follows: ∫π(2x)² - (x²)² dx. This would give us the total volume of the solid in cubic units.

## 4. How does the washer method differ from the disk method?

The washer method and disk method are both used to find the volume of solids of revolution, but they differ in the shape of the slices used. The disk method uses circular slices, while the washer method uses annular or ring-shaped slices. The washer method is more versatile as it can be used for solids with holes or irregular shapes.

## 5. What are some tips for sketching the volume using the washer method?

When sketching the volume using the washer method, it is important to first identify the axis of rotation and the boundaries of the solid. Then, you can sketch the washers and label the variables and expressions used in the integral. It is also helpful to label the axes and include units for the volume. Additionally, you can use computer software or graphing calculators to help visualize the solid and check your work.

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