Integral and cylindrical shell

In summary: If it was the lower part, you still could use cylindrical shells, right?Actually, x = 2 is a bound because 0 <= x <= 2. Yeah, the book gives the area for the top part...but I don't see why it can't be the lower part.I think I see where you are getting confused...there is a difference between what x varies over, and what the bounds are.In this case x varies from 0 to 2. That is different from a bound. Suppose the curve y=x^2 was bounded by y=0, and x=1, but x varied from 0 to a billion? It doesn't much matter how x varies so long as the
  • #1
merced
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  • #2
merced said:
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

y = [tex]x^2[/tex]
y = 4
x = 0

[tex] 0\leq x\leq 2 [/tex]

So I drew these:
http://img131.imageshack.us/img131/9225/math28qi.th.jpg [Broken]
http://img68.imageshack.us/img68/866/math21ow.th.jpg [Broken]

I don't know how to determine which region to use (above or below
y =[tex]x^2[/tex]). Either seems like it will work.

Think (as the question advised) in terms of cylindrical shells (elements). You are rotating about the y-axis. What is the radius of the cylinder? What is the height? Hence determine the volume of that element, and set up the integral.
 
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  • #3
merced said:
I don't know how to determine which region to use (above or below
y =[tex]x^2[/tex]). Either seems like it will work.

The region above y=x^2 is sort of in the shape of a bullet pointed straight down the y-axis, and the region below y=x^2 forms a sort of dog dish shaped object. Notice that x=2 is not a bound for the curve. Look at what does bound the curve and that is what you need to find the volume of.
 
  • #4
Actually, x = 2 is a bound because 0 <= x <= 2. Yeah, the book gives the area for the top part...but I don't see why it can't be the lower part.

If it was the lower part, you still could use cylindrical shells, right?
 
  • #5
merced said:
Actually, x = 2 is a bound because 0 <= x <= 2. Yeah, the book gives the area for the top part...but I don't see why it can't be the lower part.
I think I see where you are getting confused...there is a difference between what x varies over, and what the bounds are.

In this case x varies from 0 to 2. That is different from a bound. Suppose the curve y=x^2 was bounded by y=0, and x=1, but x varied from 0 to a billion? It doesn't much matter how x varies so long as the problem makes sense. In this case, x=2 is not a bound, in which case you have the curve y=x^2, y=4 and x=0 (y-axis is x=0) as bounds. Draw that region, then rotate it about the y-axis and you have your volume. Set up the integral and evaluate.
 

What is an integral shell?

An integral shell is a type of shell structure that is created by rotating a curve or an area around a central axis. It is also known as a revolving or rotational shell.

What is a cylindrical shell?

A cylindrical shell is a type of integral shell that is created by rotating a rectangle around its longer side. It is a common shape used in engineering and construction, especially for storage containers and pipes.

What are the advantages of using integral and cylindrical shells in construction?

Integral and cylindrical shells have several advantages in construction, including their high strength-to-weight ratio, ease of construction, and ability to span large distances without the need for internal supports. They also have a visually pleasing curved shape and can withstand high amounts of internal or external pressure.

What are the limitations of using integral and cylindrical shells?

One limitation of using integral and cylindrical shells is their vulnerability to buckling, especially if the shell is thin. They may also be more expensive to construct compared to other types of structures and may require skilled labor for their construction.

How are integral and cylindrical shells used in real-world applications?

Integral and cylindrical shells have a wide range of applications in various industries, including aerospace, automotive, and construction. They are commonly used for storage tanks, pressure vessels, and pipes, as well as in the design of airplane and car bodies. They are also popular in architectural design for their aesthetic appeal.

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