Find volume using disk/washer/shell method

In summary, to find the volume of the solid generated by revolving the region bounded by the equations y=x, y=0, y=4, x=5 about the line x=5, the washer method can be used with the formula V=pi∫R(y)^2-r(y)^2 dy from y=0 to y=4. The equations for R and r can be determined by drawing a sketch of the region and calculating the distances from the line x=5. The large radius, R, is 5-y and the small radius, r, is 0 because when rotating around the line x=5, there is no inner radius.
  • #1
kari82
37
0
Find volume of the solid generated revolving the region bounded by the graphs of the equations about line x=5

y=x, y=0, y=4, x=5

My plan is to use washer method. V=pi∫R(y)^2-r(y)^2 dy from y=0 and y=4

Im having trouble finding the equations for R and r. Can someone please explain me what would be a way to find those equations? Thanks!
 
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  • #2
kari82 said:
Find volume of the solid generated revolving the region bounded by the graphs of the equations about line x=5

y=x, y=0, y=4, x=5

My plan is to use washer method. V=pi∫R(y)^2-r(y)^2 dy from y=0 and y=4

Im having trouble finding the equations for R and r. Can someone please explain me what would be a way to find those equations? Thanks!

Have you drawn a sketch of the region being revolved? The typical area element is a trapezoid whose left edge is along the line y = x and whose right edge is along the line x = 5.
 
  • #3
yes, i did... and the book says that R=5-y and r=0.. but I can't see why r=0..
 
  • #4
I think I got it.. r=0 because when it rotates along x=5 there is no inner radius?
 
  • #5
Both radii are calculated as distances from the line x = 5. The large radius, R, is 5 - y. The small radius is 5 - 5 = 0. IOW, all of the disks have a radius of 5 - y. Does that make sense?
 
  • #6
Yes! Thank you so much!
 

1. What is the disk/washer/shell method used for in finding volume?

The disk/washer/shell method is a mathematical technique used to find the volume of a solid of revolution, which is a 3-dimensional shape formed by rotating a 2-dimensional shape around an axis. It is commonly used in calculus to solve problems involving finding the volume of objects with curved surfaces, such as spheres, cones, and cylinders.

2. What is the difference between the disk, washer, and shell method?

The disk method is used when the cross-section of the solid is a disk, the washer method is used when the cross-section is a washer (a disk with a hole in the middle), and the shell method is used when the cross-section is a cylindrical shell. These methods differ in the way the shape is sliced and stacked to find the volume.

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

The integral for the disk method is ∫π(r(x))^2dx, where r(x) is the radius of the cross-section at a given point. For the washer method, the integral is ∫π(R(x)^2 - r(x)^2)dx, where R(x) is the outer radius and r(x) is the inner radius. For the shell method, the integral is ∫2πrh(x)dx, where r is the distance from the axis of rotation and h(x) is the height of the cylindrical shell at a given point.

4. When should I use the disk/washer/shell method instead of other methods?

The disk/washer/shell method is most useful when the solid has a circular or cylindrical shape, and when the axis of rotation is either the x-axis or the y-axis. If the shape has a different cross-section or the axis of rotation is not aligned with the x or y-axis, other methods such as the cross-sectional area method or the cylindrical shell method may be more appropriate.

5. Can the disk/washer/shell method be used to find volumes of more complex shapes?

Yes, the disk/washer/shell method can be extended to find the volume of more complex shapes by breaking them down into smaller, simpler shapes. This is known as the "method of slicing" or the "method of washers and shells." By adding up the volumes of these smaller shapes, we can find the volume of the entire object.

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