# Find the Volume (Solid of Revolution)

• PhysicsLover0
In summary, the friend is trying to find the volume of a solid revolving around the x-axis, and they are stuck because they do not have the entire problem. They need to give us the region that will be revolved around, and they need to show what they have tried. They also need to provide at least one graph so that we can see what they are talking about.
PhysicsLover0
Find the volume of the solid y = sinx, y=cosx, and x= pi/4, revolving around x-axis
I didn't really get this at all... do I plug pi/4 for x in y=sinx, y=cosx to get the integra boundaries?

Last edited:
You need to give us the entire problem, including the axis this region has been revolved around.

After that, show us what you have tried. If you're stuck, your book most likely has some similar examples that show how to calculate volumes of revolution by cylindrical shells or by washers.

Mark44 said:
You need to give us the entire problem, including the axis this region has been revolved around.

After that, show us what you have tried. If you're stuck, your book most likely has some similar examples that show how to calculate volumes of revolution by cylindrical shells or by washers.

Sorry, forgot to mention it's revolving around the x-axis. Anyways, I got this from a friend pi * Int[0,pi/4] (cosx)^2 - (sinx)^2 dx), and I was wondering where the lower boundary, 0, came from.

The description of the region that will be revolved around the x-axis is not as complete as I would like. I believe it is the region bounded by the graphs of y = sin x and y = cos x between x = 0 (the y-axis) and the line x = pi/4.

If this is the right description, the region is sort of triangular, but with two curved sides. The "base" of this region runs along the y-axis between 0 and 1, and the two curves intersect at (pi/4, sqrt(2)/2).

You should have at least one graph: one showing the region to be revolved, and ideally, another that shows a cross-section of the volume of revolution. Your friend is using washers - disks with a hole in the middle.

The area of a washer is pi*(R^2 - r^2), where R is the larger radius and r is the smaller radius.

The volume of a washer is the area time the thickness, which is pi*(R^2 - r^2)*thickness, which can be either dx (vertical washers) or dy (horizontal washers).

For the limits of integration, figure out where the washers run. Vertical washers run left to right along the interval in question. Horizontal washers run bottom to top along the interval in question.

Can you get started with that?

## 1. How do I find the volume of a solid of revolution?

To find the volume of a solid of revolution, you need to use the formula V = π∫ba(f(x))2 dx, where a and b are the limits of integration and f(x) is the function that forms the shape of the solid when rotated around the axis of revolution.

## 2. What is the axis of revolution?

The axis of revolution is the line or axis around which the shape is rotated to form the solid. It can be any line, such as the x-axis or y-axis, depending on the problem.

## 3. How do I determine the limits of integration?

The limits of integration are the boundaries within which the function is being rotated. These can be determined by looking at the graph of the function and identifying the points where the shape begins and ends.

## 4. Can I use this method to find the volume of any solid?

No, this method can only be used for solids that are formed by rotating a 2-dimensional shape around an axis of revolution. For other types of solids, different methods must be used.

## 5. Are there any alternative methods for finding the volume of a solid of revolution?

Yes, there are other methods such as the disk method and the shell method. These methods involve slicing the solid into thin disks or shells and summing up their volumes to find the total volume of the solid. The method used will depend on the shape of the solid and the given information.

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