# Method of shells around a different axis

• Jbreezy
In summary, the conversation discusses using the method of cylindrical shells to find the volume generated by rotating a region bounded by given curves about a specified axis. The equations y = x^2 and y = 2-x^2 are used, and the attempt at a solution involves finding the height and radius and using the formula 2∏rhΔr to calculate the volume. The final answer obtained is 16∏/3, which is confirmed to be correct by comparing it to the result obtained using the method of disks.
Jbreezy

## Homework Statement

Use the method of cylindrical shells to find the volume generated by rotation the region bounded by the given curves about the specified axis.

## Homework Equations

y = x^2, y = 2-x^2; about x = 1

## The Attempt at a Solution

I tried to just break it down.
I want something of the form 2∏rhΔr
OK so To find the height f(x) I subtracted.
2-x^2-x^2 = 2-2x^2. For the radius I did a-x so 1-x is the radius

So I have

V = 2∏∫ (1-x)(2-2x^2)dx between -1 and 1 because that is where the graphs intersect.
Evaluating it I got 2∏((x^4)/2 -(2x^3)/3 -x^2 +2x ] between -1 and 1

I got 16∏/3
Is this the right way?

Yes, it is correct. Try doing it using disks and see if you can obtain the same answer. This will allow you to compare the complexity of the resulting integrals and see why one method is more efficient than the other in this case.

## 1. What is the method of shells around a different axis?

The method of shells around a different axis is a mathematical technique used to calculate the volume of a solid of revolution by slicing it into thin cylindrical shells and integrating their volumes.

## 2. How does the method of shells around a different axis differ from the method of disks?

While both methods involve slicing a solid of revolution into thin pieces, the method of shells uses cylindrical shells that are stacked around a different axis, while the method of disks uses circular disks stacked perpendicular to the axis of revolution.

## 3. What types of shapes can the method of shells around a different axis be applied to?

The method of shells can be applied to any solid of revolution, including cones, spheres, and objects with irregular shapes.

## 4. What is the formula for calculating the volume of a solid using the method of shells around a different axis?

The formula is: V = ∫2πxf(x)dx, where x is the distance from the axis of revolution to the shell, and f(x) is the function that represents the cross-sectional area of the solid at that distance.

## 5. What are some real-world applications of the method of shells around a different axis?

The method of shells is commonly used in engineering and physics to calculate the volume of objects with rotational symmetry, such as water tanks, pipes, and turbines. It can also be applied in calculus to solve problems involving rotational motion and moments of inertia.

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