# Sketch & Find Volume of Rotated Shell: y=x^2+1, y=-1

• bonfire09
In summary, the task is to sketch a strip in either a vertical or horizontal orientation and state an expression for the volume of the shell formed by rotating it around the line y=-1. The given equations are y=x^2+1, x=0, y=1, and x=1. The student is unsure of how to get started and is considering using dy for the integral. They have been asked to share their progress with the integral to receive further assistance.
bonfire09

## Homework Statement

I have to sketch a vertical or horizontal strip and state an expression for the volume of the shell formed by its rotation. Do not solve

y=x^2+1 x=0
y=1 x=1

around y=-1

## The Attempt at a Solution

I'm really lost because I don't know how to get started. I think dy works but I can't get it right. Any help will be great thanks.

Have you figured out the shape in the plane? As a volume of revolution?
Pls post what you have for the integral so that we can see where you are stuck.

## 1. What is the process for finding the volume of a rotated shell using the equations y=x^2+1 and y=-1?

To find the volume of a rotated shell using these equations, you first need to set up an integral that represents the volume. This can be done by taking the integral of the cross-sectional area of the shell, which is given by π(y)^2. Then, you need to set up the limits of integration by finding the points of intersection between the two equations. Finally, integrate the expression over these limits to find the volume.

## 2. Can I use any other equations to find the volume of a rotated shell?

Yes, you can use any equations that represent the shape of the shell to find its volume. However, the equations need to be in terms of the variable of rotation, which in this case is y. You can also use equations that are rotated about the x-axis, in which case the variable of rotation would be x.

## 3. What is the significance of the equation y=x^2+1 in finding the volume of a rotated shell?

This equation represents the curve of the shell, which is rotated around the y-axis to form the three-dimensional shape. It is important to know the equation of the shell in order to set up the integral for finding the volume.

## 4. Can I find the volume of a rotated shell using the equation y=x^3+1 instead?

Yes, you can use this equation to find the volume of a rotated shell. However, the process for setting up the integral and finding the limits of integration may be different since the shape of the shell would be different.

## 5. Is there a limit to the number of equations I can use to find the volume of a rotated shell?

No, there is no limit to the number of equations that can be used. However, it is important to ensure that the equations represent the shape of the shell accurately and that the integral is set up correctly to find the volume.

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