Calculating Volume of Region Revolved About y=4

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In summary, to calculate the volume of a region revolved about y=4, you can use the disk method or the shell method by integrating the cross-sectional area of the region. The disk method involves integrating the area of thin disks perpendicular to the axis of revolution, while the shell method involves integrating the volume of thin cylindrical shells parallel to the axis of revolution. The axis of revolution must be a horizontal or vertical line, and the limits of integration for the disk or shell method depend on the orientation of the region and the axis of revolution. These methods can only be used for regions that can be rotated around a horizontal or vertical axis, with a clear boundary and no holes or gaps. Other methods must be used for regions that do not meet
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bard
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Find the region enclosed by x=3y and x=-y^2+4. Set up integrals both shell and disc that represent the volume when this region is revolved about y=4.

i got the shell method, how would i represent the disc method>

pi integral(-4,1) (3y-4)^2-(-y^2)^2 --this is wat i got for disc method, though i am sure it is wrong...
 
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It might help you if you make a drawing -- draw the line that you're rotating about, and the region that is being rotated.
 

1. How do I calculate the volume of a region revolved about y=4?

To calculate the volume of a region revolved about y=4, you will need to use the disk method or the shell method. Both methods involve integrating the cross-sectional area of the region with respect to the axis of revolution.

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

The disk method involves slicing the region into thin disks perpendicular to the axis of revolution and integrating the area of each disk. The shell method involves slicing the region into thin cylindrical shells parallel to the axis of revolution and integrating the volume of each shell.

3. Can I use any axis of revolution when calculating volume?

No, the axis of revolution must be a horizontal or vertical line. If the axis is not parallel to the x or y-axis, you will need to use the washer method to calculate the volume.

4. How do I determine the limits of integration when using the disk or shell method?

The limits of integration for the disk or shell method depend on the orientation of the region and the axis of revolution. When using the disk method, the limits will be the x-values of the points of intersection between the region and the axis of revolution. When using the shell method, the limits will be the y-values of the points of intersection.

5. Can I use the disk and shell methods to calculate the volume of any region?

No, the disk and shell methods can only be used for regions that can be rotated around a horizontal or vertical axis. Additionally, the region must have a clear boundary and cannot contain holes or gaps. If the region does not meet these criteria, you will need to use other methods such as the method of cylindrical shells or the method of slices.

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