Finding Volumes by using the Disc and Washer Method

In summary, the disc and washer method involves finding the volume of a solid by integrating the cross-sectional areas of discs or washers. The outer and inner radius are determined by the functions that form the boundaries of the region being integrated. In the example given, the functions f(x)=sec x and g(x)=tan x represent the boundaries x=0 and x=1, respectively, and help to define the region. By graphing the functions, it can be determined which function represents the outer and inner radius.
  • #1
scrage
2
0
Theres a few key concepts about the disc and washer method that I can't quite grasp and I was hoping if I could get a bit of clarification.

1) How do you find your outer and inner radius? I can provide an example if needed.

2) If a problem has its function, for example f(x)= sec x, g(x)=tan x, x=0, and x=1.. what do the values x=0 and x=1 mean in the question. Do they represent some sort of asymptote?

Thanks,
David
 
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  • #2
The [itex]x=0,x=1[/itex] are also bounding functions for your region of interest.

The best way to determine bounds for these things is the actually draw the picture (grpah) of what is going on...

In this instance, the region is a vaguely rectangular thing with [itex]x=0[/itex] on the left, [itex]x=1[/itex] on the right, [itex]\sec(x)[/itex] on top, and [itex]\tan(x)[/itex] on bottom - this is all viewing it basically with respect to the [itex]x[/itex] variable (in the traditional way).

Viewing it with respect to the [itex]y[/itex] variable (on it's side), then it is much messier. I'll leave that to you.
 
  • #3
scrage said:
Theres a few key concepts about the disc and washer method that I can't quite grasp and I was hoping if I could get a bit of clarification.

1) How do you find your outer and inner radius? I can provide an example if needed.

2) If a problem has its function, for example f(x)= sec x, g(x)=tan x, x=0, and x=1.. what do the values x=0 and x=1 mean in the question. Do they represent some sort of asymptote?
Did you try drawing the graphs, y= sec(x), y= tan(x), y= 0, and y= 1? They are the boundaries of the region referred to.

Thanks,
David
 
  • #4
Yes, after graphing it I can see that the area inbetween x=0 and x=1 is the "area" wanted to find the volume. For the radii, would f(x)=sec x be the outer (or upper)? and g(x)=tan x the inner (or lower)?
 
  • #5
It should be evident from your graph. Which of those two functions is greater in the interval 0<x<1?
 

1. What is the Disc and Washer Method for finding volumes?

The Disc and Washer Method is a mathematical technique used to find volumes of three-dimensional shapes by slicing the shape into thin discs or washers and integrating them to find the total volume.

2. When is the Disc and Washer Method typically used?

The Disc and Washer Method is typically used when the shape being measured has a circular cross-section and is revolved around a specific axis. This method is commonly used in calculus to find the volume of solids of revolution.

3. How do you set up the integral for the Disc and Washer Method?

The integral for the Disc and Washer Method is set up by first determining the limits of integration, which are the values of x where the shape begins and ends. Then, the integrand is determined by taking the cross-sectional area of the shape and integrating it with respect to x or y, depending on the axis of rotation.

4. What are the differences between the Disc Method and Washer Method?

The Disc Method is used when the shape is being revolved around an axis that passes through the cross-section, while the Washer Method is used when the axis of rotation does not pass through the cross-section. Additionally, the Washer Method uses two integrals, one for the outer radius and one for the inner radius, while the Disc Method only requires one integral.

5. Can the Disc and Washer Method be used for non-circular cross-sections?

No, the Disc and Washer Method is only applicable to shapes with circular cross-sections. For non-circular shapes, other methods such as the Shell Method or Cross-Sectional Area Method must be used to find the volume.

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