Volume of Rotated Region Using Cylindrical Shells - Step-by-Step Guide

In summary, the conversation is about finding the volume generated by rotating a region bounded by given curves about a specified axis, using the method of cylindrical shells. The person is asking for help with setting up the integral, and has provided their attempted solution, which is incorrect. The other person explains the shape of each shell and how to find its area, leading to the correct answer of 24pi.
  • #1
shaneewert
2
0
Please help! How do I do this problem?

Using the method of cylindrical shells, find the volume generated by rotating the region the region bounded by the given curves about the specified axis.

y=(x-1)^(1/2), y=0, x=5; about y = 3

Please tell me how to set up the integral! Any help is MUCH appreciated.

So far I have Integral from 0 to 2 of (3-(y^2+1))*y dy I know that isn't right, because I am not getting the right answer! The book says that it is 24pi.

Thank you.
 
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  • #2
Do you know what each shell looks like?

The cross section of each shell is a long, narrow, flat rectangle, whose long dimension is parallel to the x-axis and stretches from the point where x = y^2 + 1 to the point where x = 5. It is parametrized by y. The shell itself is that rectangle rotated around the line y = 3. Ignore the small vertical width of the rectangle for a minute (which is dy) so that the shell is just like a horizontal tube with no thickness. What is the area of the outer surface of this tube? It has a radius--the radius is 3 - (y^2 + 1). So what is its circumference? What is its length?
 

FAQ: Volume of Rotated Region Using Cylindrical Shells - Step-by-Step Guide

What is the formula for finding the volume of a rotated region using cylindrical shells?

The formula for finding the volume of a rotated region using cylindrical shells is V = 2π∫(radius)(height)(thickness)dx, where the radius is the distance from the axis of rotation to the edge of the shell, the height is the difference between the upper and lower boundaries of the region, and the thickness is the width of the shell. This formula is derived from the formula for the volume of a cylinder (V = πr^2h) and is used to find the volume of the entire rotated region by adding up the volumes of all the cylindrical shells.

How do I determine the limits of integration for finding the volume of a rotated region using cylindrical shells?

The limits of integration for finding the volume of a rotated region using cylindrical shells are determined by the boundaries of the region being rotated. These boundaries will become the upper and lower limits of the integral. To determine these limits, you can either graph the region and use the x-values of the intersections with the axis of rotation, or you can set up an equation with the boundaries and solve for the x-values.

Can the method of cylindrical shells be used to find the volume of a rotated region with a hole?

Yes, the method of cylindrical shells can still be used to find the volume of a rotated region with a hole. You will need to subtract the volume of the hole from the total volume of the region. To do this, you can either find the volume of the hole using the formula for the volume of a cylinder and subtract it from the total volume, or you can set up a separate integral for the hole and subtract it from the integral for the entire region.

What is the difference between using the method of cylindrical shells and the method of washers to find the volume of a rotated region?

The main difference between using the method of cylindrical shells and the method of washers to find the volume of a rotated region is the shape of the cross-section used in the integral. The method of cylindrical shells uses cylindrical shells as the cross-section, while the method of washers uses disks or washers. The choice of method depends on the shape of the region and which method will be easier to set up and integrate.

Can the method of cylindrical shells be used to find the volume of a non-rotated region?

No, the method of cylindrical shells can only be used to find the volume of a region that is rotated around an axis. If the region is not rotated, you will need to use a different method, such as the method of cross-sections or the method of slices, to find the volume. These methods involve slicing the region into thin slices and using the formula for the volume of a shape (such as a rectangle or triangle) to find the volume of each slice, then adding them up to find the total volume.

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