Volume, washer method and shell method

In summary, the conversation discusses the concept of rotating a solid formed by a bounded area around an axis. This can be done by slicing the solid into thin circles or "washer" shapes and calculating their volumes. The equations for the washer method involve taking the integral of the function squared and accounting for the different orientations of the slices. The conversation also includes a helpful tip for determining which variable to integrate over.
  • #1
JKLM
21
0
I don't understand how to set up the washer and shell equations. When you are given the function and the line to rotate it around, or two functions and a line.
 
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  • #2
Suppose we are given an area bounded by certain curves in the xy-plane. That area is rotated around an axis to form a solid. Each point of the figure is rotated in a circle. If you slice the solid perpendicular to the axis you will see a "cross section" (the area revealed by the slice) that is either a full disk or a "washer" (the area between two circles. It's easy to calculate the area of a circle and you can imagine the solid as made of a lot of very thin circles.

For example, suppose the part of the parabola y= x2 for 0< x< 1 is rotated around the y-axis. You get a parabolic solid and if you slice through it perpendicular to the y-axis, at any y, the "cut end" is a circle.

In fact, imagine that this figure is a potato! Slice it into very thin slices to make potato chips. Each slice is a circle (disk, more properly). Its area is &pi;r2 and, taking "h" to be its thickness, its volume is &pi;r2h.

If we put the slices back together we could reform the potato and the volume of the potato is the sum of the volumes of the slices.

Of course, "r", the radius of the circle, varies from slice to slice. If our potato were really shaped like y= x2 rotated around the y-axis, then, since we are slicing perpendicular to the y-axis, "r" is equal to the x value: x= &radic;(y).

The area of the slice is then &pi;r2= &pi;y and, setting this up as a "Riemann sum", the thickness, because it is measured along the y-axis, is dy: the volume of each slice is &pi;y dy and the sum of all the slices (as we imagine the slices becoming infinitesmally thin) becomes the integral of &pi;y dy from y=0 to 1:
(1/2)&pi;y2 (evaluated between 0 and 1)= &pi;/2.
 
  • #3
i understand the theory behind the washer method, but what i don't get is the application of the theory, i don't understand how to set up the equations
 
  • #4
With the washer/disc method, if you're given a function f(x) and you want to rotate it, say around the x-axis - then you simply take a representative rectangle with its base on the x-axis (and its length going up to the function).

Then the width/base of the representative rectangle will be dx, while the length will be f(x).

When you rotate the function about the x-axis, that rectangle will essentially become the radius of the circular disc. So,

[tex]V=\pi\int_{a}^{b}f(x)^2dx[/tex]

For something with two functions, just take:

[tex]V=\pi\int_{a}^{b}f(x)^2-g(x)^2dx[/tex]

Where f(x) is a function above g(x).

These are the steps I use to set up the equations for the disc method. Does that help?
 
  • #5

explains everything
 
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  • #6
this helps me when writing the integral... hey vex... if the drawn washer/disk is vertical (goes up and down) integrate over x (use dx), if it is horizontal (goes left and right) integrate over y (use dy)
 

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

The washer method and the shell method are both techniques used to calculate the volume of a solid of revolution. The difference between the two lies in the shape of the cross-section used to create the solid. The washer method uses circular cross-sections, while the shell method uses cylindrical shells.

How do you determine which method to use?

The choice between using the washer method or the shell method depends on the shape of the solid and the axis of revolution. If the solid has a circular base and the axis of revolution is perpendicular to the base, the washer method is usually used. If the solid has a non-circular base or the axis of revolution is parallel to the base, the shell method is typically used.

What is the formula for calculating volume using the washer method?

The formula for calculating volume using the washer method is V = π∫a2-b2dx, where a and b represent the outer and inner radii of the cross-section, and dx represents the width of the cross-section. This integral is typically evaluated over the interval of integration that corresponds to the axis of revolution.

What is the formula for calculating volume using the shell method?

The formula for calculating volume using the shell method is V = 2π∫r(x)h(x)dx, where r(x) represents the distance from the axis of revolution to the outer edge of the shell and h(x) represents the height of the shell. This integral is typically evaluated over the interval of integration that corresponds to the axis of revolution.

Can the washer and shell methods be used for solids with holes?

Yes, both the washer method and the shell method can be used to calculate the volume of solids with holes. In these cases, the formula for calculating volume may need to be adjusted to account for the hole or holes in the solid. This can be done by subtracting the volume of the hole(s) from the total volume calculated using the respective method.

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