Volume of Rotated Function: Shell Method for y=1/x on [1,4]

In summary, the problem involves finding the volume of a function rotated about the x-axis over the interval [1,4]. The approach is to sketch the region and express it as a sum of two integrals. One integral is for the region between y=0 and y=1/4, where the length of the shell is constant at 3. The other integral is for the region between y=1/4 and y=1, where the radius is y and the length of the shell depends on y. The shell method is used, which involves integrating 2*pi*r*l, with r as the radius of the shell and l as the length of the shell.
  • #1
physstudent1
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Homework Statement


Find the volume of the function rotated about the x-axis over the given interval:

y=1/x

on interval [1,4]

for this specific problem it says to sketch the region and express as a sum of two integrals.

Homework Equations


The Attempt at a Solution



I'm not sure why you need to express it as a sum of two integrals. Since its about the x-axis I solved so that it was x=1/y for the radius and the height would then be y therefore with the shell method it would be 2*pi* [tex]\int[/tex] (1/y)y(dy) the limits of integration being 1 to 4
 
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  • #2
Shells means integrating 2*pi*r*l where r is the radius of the shell and l is the length of the shell. There are two regions of integration because between y=1/4 and y=1 the radius is y and the length depends on y (but it's not what you wrote). Between y=0 and y=1/4 the length of the shell is constant and equal to 3. If you sketched the region (and you should) I would suggest using that sketch.
 
  • #3
Ohhhh, okay I get what your saying about the the 2 shells I see on the graph now as well, however I am confused about how to find what the radius is in terms of y I thought you just solved the original equation so that it was x= ??
 

FAQ: Volume of Rotated Function: Shell Method for y=1/x on [1,4]

1. What is the Shell Method for finding the volume of a rotated function?

The Shell Method is a mathematical technique used to find the volume of a solid shape that is formed by rotating a function around a given axis. It involves integrating the product of the circumference of a shell, the height of the shell, and the thickness of the shell.

2. How do you apply the Shell Method to find the volume of y=1/x on [1,4]?

To apply the Shell Method to find the volume of y=1/x on [1,4], we first need to determine the axis of rotation. In this case, the function y=1/x is being rotated around the x-axis. Next, we need to set up the integral, which would be ∫(2πx)(1/x)dx, where x represents the thickness of the shell. Finally, we evaluate the integral using the limits of integration, in this case, from 1 to 4.

3. What are the advantages of using the Shell Method over the Disk Method for finding volume?

The Shell Method has several advantages over the Disk Method. Firstly, it can be applied to functions that are not easily expressed in terms of y, as the Disk Method requires. Secondly, it is more efficient for functions that are difficult to integrate. Lastly, it can be used to find the volume of more complex shapes, such as those with holes or voids.

4. Can the Shell Method be used for finding the volume of a 3-dimensional shape?

Yes, the Shell Method can be used for finding the volume of a 3-dimensional shape. However, it is important to note that the shape must have a circular cross-section in order for the Shell Method to be applicable. If the shape does not have a circular cross-section, other techniques such as the Washer Method or Cylindrical Shells Method may need to be used.

5. Are there any limitations to using the Shell Method for finding volume?

The Shell Method does have some limitations. As mentioned earlier, it can only be used for shapes with circular cross-sections. Additionally, it may not be applicable for functions that have discontinuities or sharp corners, as it relies on integration. In such cases, other techniques may need to be used to find the volume of the shape.

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