Volume of Region Rotated about X-Axis Using Shell Method

So you're going to need to use two integrals.In summary, to find the volume of the region rotated about the x-axis, the shell method is used. Two integrals from 0 to 25 are needed, with the boundary function changing at (-2, 4). The attempt at a solution provided an integral, but it did not account for the changing boundary function. This may explain why the resulting answer is incorrect.
  • #1
whatlifeforme
219
0

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 ]
 
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  • #2
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 ]

What's the question?
 
  • #3
is my attempt correct so far? just setting up the integral? becuase i keep getting the wrong answer.

my answer: 1111 (pi); correct answer: 5488pi/5
 
  • #4
First off, did you draw a picture of the region, and did you draw a sketch of the solid of revolution?

If so, you should have noticed that one integral isn't going to work for this problem. The boundary function on the left changes at the point (-2, 4).
 

What is the Shell Method?

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.

How is the Shell Method different from the Disk Method?

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.

What is the formula for using the Shell Method?

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.

How is the Shell Method applied to finding the volume of a region rotated about the x-axis?

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.

What are some common mistakes to avoid when using the Shell Method?

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.

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