# Simpsons rule volume washers shells about y=-1

• z37002
In summary, the conversation discusses using Simpsons rule with n=6 to approximate the volume of a solid obtained by rotating the region bounded by y=x^3, y=1, x=0 about the line y=-1. The question is asked about which volume integral to use and how to put it into Simpsons rule form. The conversation also mentions the difficulty of combining Simpsons rule with shells when rotating about a line rather than an axis. The advice given is to shift the origin and align one of the principle axes with the rotation axis to simplify the problem.
z37002
Simpsons rule n=6, to approximate volume of solid obtained by rotating the region bounded by y=x^3,y=1,x=0 about line y=-1. What is the volume integral to use, and how is it put into simpsons rule form? thanks.

Welcome to PF;
Those are questions for you ... how have you attempted to answer them so far?
Hint:
State Simpson's rule
Sketch the region
Shift the origin

Have you studied solids of rotation before?

Yes, I've done both revolution of solid and simpsons, but not together? And it is the about y=-1 as opposed to on the axis that I seek help w. And if the integral I know is shells, than how do plug in x values if integral is in y's? I am not seeing this as being able to be simple discs. Though a shift of y-axis shouldn't effect x values, I still see the y=-1 as a deterrent to combining simpsons sand shells. IF another way or similar example please share.

z37002 said:

You know how to do solids of rotation about one of the principle axes of the coordinate system.
However, you don't know how to handle arbitrary rotations?

What you have to do is realize that the positon and orientation of the axes doesn't matter for the problem - so move to a new coordinate system where one of the principle axes lines up with the rotation axis.

You should see how to do this if you sketch the situation out.

## 1. What is Simpsons rule and how is it used to calculate volume?

Simpsons rule is a numerical integration method used to approximate the area under a curve. In the context of calculating volume, it involves dividing the region into smaller sections and using the formula V = π∫(R^2 - r^2)dx, where R is the outer radius of the region, r is the inner radius, and dx represents the width of each section. By summing the volumes of these sections, an estimate of the total volume can be obtained.

## 2. How does Simpson's rule apply to calculating volume for washers and shells?

When using Simpson's rule to calculate volume for washers and shells, the region is divided into vertical sections (shells) or horizontal sections (washers). The inner and outer radii are then determined for each section and plugged into the formula V = π∫(R^2 - r^2)dx. The resulting sum provides an estimate of the total volume.

## 3. What is the significance of the y=-1 in this problem?

The y=-1 in this problem represents the line of symmetry for the region being calculated. This means that the region will be divided into equal sections above and below the line, simplifying the calculation process.

## 4. Is Simpson's rule the most accurate method for calculating volume?

Simpson's rule is a relatively accurate method for calculating volume, but it is not always the most accurate. It can produce more accurate results for regions with curved boundaries compared to other methods such as the disk method or the cylindrical shell method. However, for more complex regions, other methods may be more accurate.

## 5. How can I check the accuracy of my calculated volume using Simpson's rule?

One way to check the accuracy of your calculated volume using Simpson's rule is to divide the region into smaller sections and recalculate the volume. If the results are similar, then the initial calculation is likely accurate. Additionally, you can also compare the results to the volume calculated using other methods to ensure consistency.

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