Integration: Washer and shell method

In summary, the conversation discusses setting up an equation to find the volume of a shape rotated about the axis of x = 10. The equation y = 4-e^x is given and converted to x = ln(4-y), with values for R(y) and r(y) also provided. The conversation then mentions using the disk and shell methods to calculate the volume, with the correct equation being v = 2pi integrate from 0 to ln4 (10-x)(4-e^x)dx. Finally, it is revealed that the individual has figured out the solution.
  • #1
Bryon
99
0
Hi Everyone

I am having a trouble setting up an equation: I am to find the volume by integrating the equation y = 4-e^x from 0 to 3 and rotate it about the axis of x = 10.

Here is what I have:

the equation y= 4-e^x in terms of x is x = ln(4-y)


R(y) = 10
r(y) = ln(4-y)

the equation I have using the disk method is this:

v = 2pi integrate from 0 to 3 (10^2)-(ln(4-y))^2)dy

using the shell method I have (which I think is correct):

v = 2pi integrate from 0 to ln4 (10-x)(4-e^x)dx
 
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  • #2
Wait...I figured it out. :biggrin:
 

FAQ: Integration: Washer and shell method

How do you determine when to use the washer method and when to use the shell method for integration?

The washer method is used when the cross section of the solid being rotated around the axis of integration is a disk or a washer shape. The shell method is used when the cross section is a cylindrical shell. You can determine which method to use by examining the shape of the cross section.

What is the difference between the washer method and the shell method for integration?

The main difference between the two methods is the shape of the cross section. The washer method uses disks or washers, while the shell method uses cylindrical shells. The washer method also requires integration in the form of "top-bottom" while the shell method requires integration in the form of "left-right".

How do you set up the integral for the washer method?

To set up the integral for the washer method, you need to define the inner and outer radii of the washers, and the limits of integration. The inner radius is the distance from the axis of rotation to the inner edge of the cross section, and the outer radius is the distance to the outer edge. The limits of integration are determined by the boundaries of the region being rotated around the axis.

Can you use both the washer method and the shell method for the same problem?

Yes, it is possible to use both methods for the same problem. However, one method may be easier to use than the other depending on the shape of the cross section and the given boundaries of the region being rotated.

What is the purpose of using the washer and shell methods for integration?

The washer and shell methods are used to find the volume of irregularly shaped objects or solids with curved surfaces. By rotating a cross section of the object around an axis of rotation, we can use integration to find the volume of the resulting solid.

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