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z37002
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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.
Post a link to your sketch.z37002 said:Any advice would be good.
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.
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.
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.
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.
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.