The Disk/Washer Method: Axis Of Revolution Question

In summary, the conversation is about finding the volume of a solid generated by revolving a region bounded by a curve and two lines around a given axis. The method used is the Disk/Washer Method, where the graph is translated to the right and the inner radius is set to 2. The reason for this translation and subtraction is to find the distance between the given points and the axis of rotation.
  • #1
carlodelmundo
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Homework Statement



http://65.98.41.146/~grindc/SCREEN01.JPG

Find the volume of the solid generated by evolving the region bounded by y = sqrt(x), y = 0, x = 4, when revolved around the line x = 6

Homework Equations



The Disk/Washer Method -

The Attempt at a Solution



let R(y) = 6 - y^2
r(y) = 2

Okay. What I don't quite understand is why we translate the graph to the right 6 units, and letting the inner radius equal 2. Can anyone shed some light on why we do this? A counter-example would help me understand this easier.

My $0.02... maybe you can clarify it:

By translating the graph y = sqrt(x), 6 units to the left, the vertex point (0,0) now becomes (6,0)... and it's similar, as if, rotated by the y-axis. I know there is a gap of 2 units (from x = 4 and x = 6, as shown by the graph). But where does the subtraction come from?

Carlo
 
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  • #2
The gap of 2 is the distance from x=6 to x=4, which is 6-4. There's a subtraction in there too. Similarly the distance from x=y^2 to x=6 is 6-y^2. Subtracting a larger number from a smaller gives the distance between them.
 

1. What is the disk/washer method for finding volume using an axis of revolution?

The disk/washer method is a technique used in calculus to find the volume of a solid of revolution, where a 2-dimensional shape is rotated around an axis to form a 3-dimensional solid. It involves taking cross-sectional slices of the solid perpendicular to the axis of revolution and using the formula V = πr^2h, where r is the radius of the slice and h is the thickness of the slice, to calculate the volume of each slice. The sum of all the individual volumes gives the total volume of the solid.

2. What is the difference between a disk and a washer in the disk/washer method?

A disk is a solid 2-dimensional shape with a circular boundary, while a washer is a solid 2-dimensional shape with two circular boundaries (one larger and one smaller) that are concentric. In the disk method, we use disks to represent the cross-sectional slices of the solid, while in the washer method, we use washers to account for the "hole" in the center of the solid.

3. Can the disk/washer method be used for any shape?

No, the disk/washer method can only be used for solids of revolution, where the shape is formed by rotating a 2-dimensional shape around an axis. The shape must also have a defined axis of revolution and the cross-sectional slices must be perpendicular to the axis.

4. How do you set up the integral for the disk/washer method?

The integral for the disk/washer method will involve integrating the formula V = πr^2h over the limits of the axis of revolution. The limits can be determined by finding the points of intersection between the shape and the axis, and the radius and thickness of the slices can be expressed in terms of the variable of integration.

5. Are there any other methods for finding volume using an axis of revolution?

Yes, there are other methods such as the shell method and the cylindrical shells method that can also be used to find the volume of a solid of revolution. These methods may be more efficient or suitable for certain shapes, so it is important to understand all the different techniques and when to use them.

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