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whatlifeforme
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Homework Statement
use the shell method to find the volume of the region rotated about the x-axis.
Homework Equations
y=3x+10
y=x^2
The Attempt at a Solution
2∏ (integral) (0 to 25) [ (y) (sqrt(y) - (y-10)/3) dy ]
whatlifeforme said:Homework Statement
use the shell method to find the volume of the region rotated about the x-axis.
Homework Equations
y=3x+10
y=x^2
The Attempt at a Solution
2∏ (integral) (0 to 25) [ (y) (sqrt(y) - (y-10)/3) dy ]
The Shell Method is a mathematical technique used to calculate the volume of a three-dimensional region that is rotated around an axis. It involves summing up the volumes of thin cylindrical shells that make up the region.
The Shell Method and the Disk Method are both techniques used to find the volume of a solid of revolution. The main difference is that the Shell Method uses cylindrical shells to approximate the volume, while the Disk Method uses circular disks. The Shell Method is often preferred for regions with "holes" or voids.
The formula for using the Shell Method is V = 2π∫[a,b]r(x)h(x)dx, where a and b are the bounds of integration, r(x) is the distance from the axis of rotation to the shell, and h(x) is the height of the shell. This formula is derived from the volume of a cylinder: V = hπr^2.
To use the Shell Method to find the volume of a region rotated about the x-axis, we first need to express the region as a function of x. Then, we can use the formula V = 2π∫[a,b]r(x)h(x)dx, where a and b are the bounds of integration and r(x) is the distance from the x-axis to the shell. We integrate from a to b because we are rotating the region around the x-axis, which is a vertical line.
One common mistake when using the Shell Method is forgetting to square the distance from the axis of rotation to the shell, which is represented by r(x). This distance should always be squared in the formula V = 2π∫[a,b]r(x)h(x)dx. Another mistake is using the wrong bounds of integration, especially when the region is not symmetrical. It is important to carefully consider the shape of the region and choose the correct bounds of integration.