Washer method and disc method for finding volumes of graphs

In summary, when finding the volume of a solid rotated on a certain axis, there are two methods that can be used: the disk method and the washer method. The disk method is used when the solid does not have a hole in the center, while the washer method is used when there is a hole in the center. Both methods involve constructing a moving volume function and finding the area of the slices. It is important to consider which method is more comfortable and makes more sense for each specific problem. It is also recommended to search for previously given explanations or to read about Cavalieri's method for a more detailed understanding of these methods.
  • #1
KataKoniK
1,347
0
Hi,

I was wondering how do you know which method to use when let's say, they give you two equations and say to find the volume of the solid rotated on a certain axis. Is there a certain rule of thumb to follow (like in stock market - buy low sell high?)? I am really confused.
Thanks in advance.
 
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  • #2
depends on how the equation are given to you.

Try this one using the disc method.

Revolve [itex]y = x^3-x[/itex] about the y-axis. You can't. Use which one is more comfortable and which one makes sense.
 
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  • #3
You could do any problem either way... But some are easier to solve using a particular method.

y=x^3 - x can be solved with disks but you have to integrate in respect to y rather than x...

Makes it a bit more complex.
 
  • #4
having answered this question in utter and exhaustive detail, months ago, I wonder, is there some way we could arrange for students to search our site for answers that have already been given?

i simply become too tired to endlessly repeat the same answers. but it seems there are usually others with more energy.
 
  • #5
Take a disk and a washer and look at them! The washer has a hole in the middle that the disk doesn't have! That's when you use the washer method- when the area rotated around the axis does not extend all the way to the axis so you will have a hole in the center of the solid.
Actually, "washer" and "disk" are basically the same method: any problem that you use the "washer" on, you could do as two "disks"- Do the "outer" disk ignoring the missing part- taking the area right up to the axis, then do the same thing finding the volume of the missing inner portion and subtract.
 
  • #6
all volumes are computed by constructing a moving volume function, which at any stage cuts out an area. the area is the derivative of the volume function, and so the problem is easier dependening on how easy the area function is. you can sweep out volumes by moving slices in any direction, upward or outward, or around a circle, or other ways. expanding a cylinder gives the shell metjhod, raising a plane along a line gives the "disc" method (assuming your volume was sewept out by a revilution), etc etc etc... i recommend you search for my earlier more detailed explanations, or simply read about cavalieri;s method.
 

What is the difference between the washer method and the disc method for finding volumes of graphs?

The washer method and the disc method are two different techniques used to find the volume of a solid of revolution. The main difference is that the washer method is used when the cross-sections of the solid are washers (or rings), while the disc method is used when the cross-sections are discs (or circles).

How do you set up the integrals for the washer method and disc method?

For the washer method, the integral is typically set up as ∫ (π * (outer radius)^2 - π * (inner radius)^2) dx, where the outer and inner radii represent the distances from the axis of rotation to the outer and inner edges of the solid, respectively. For the disc method, the integral is usually set up as ∫ π * (radius)^2 dx, where the radius represents the distance from the axis of rotation to the edge of the disc.

What are some common mistakes to avoid when using the washer method and disc method?

One common mistake is forgetting to square the radii in the integrals. Another mistake is using the wrong formula for the method (e.g. using the disc method for a solid with washers as cross-sections). It is also important to pay attention to the limits of integration, as they may differ for each method.

Can the washer method and disc method be used for any shape?

Both methods can be used for solids of revolution with circular cross-sections. However, the washer method can also be extended to other shapes, such as squares or rectangles, by using the appropriate formula for the cross-sectional area.

When should the washer method and disc method be used instead of other techniques, such as the shell method?

The washer method and disc method are most commonly used for solids of revolution with circular cross-sections. However, the shell method may be more suitable for solids with non-circular cross-sections, such as triangles or rectangles. It is important to choose the method that best fits the shape of the solid to ensure accurate results.

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