Volume of Revolution: Find V with Shell Method

In summary, the problem involves finding the volume of a container created by rotating the curve y = 0.5x^2 0 \leq x \leq 3 around x = -3 and adding a plane bottom. The shell method is used to set up an integral for the volume, taking into account the central cylinder and the circular slices in the y direction. After clarification and visualizing the problem, the correct volume is determined to be 114.75*pi.
  • #1
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Homework Statement


A container with height [tex]4.5[/tex] is created by rotating the curve [tex]y = 0.5x^2[/tex] [tex]0 \leq x \leq 3 [/tex] around [tex] x = -3 [/tex] and putting a plane bottom in the box. Find the volume [tex]V[/tex] of the box.

Homework Equations

The Attempt at a Solution


I want to solve this by using the shell method. I have put up the following integral, which will be the volume of the revolution. It is however not correct, and I haven't really used the information about the height given in the text.. could anyone help me?

[tex]2\pi\int_0^3 (x+3)(0.5x^2) \, dx[/tex]
 
Last edited:
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  • #2
You are omitting the central cylinder (between x = 0 and -6).
Personally I think it would be easier to take circular slices in the y direction.
 
  • #3
How do you go about calculating the volume of the central cylinder?
 
  • #4
I think you just edited it with the ## 2 \pi ##, and it looks correct to me.
 
  • #5
Charles Link said:
I think you just edited it with the ## 2 \pi ##, and it looks correct to me.
No, the post haven't been edited. I think it´s true what mjc123 said about omitting the central cylinder. I am however not exactly sure how to visualize it.
 
  • #6
Kqwert said:
No, the post haven't been edited. I think it´s true what mjc123 said about omitting the central cylinder. I am however not exactly sure how to visualize it.
I am a little puzzled by this one also, because it doesn't look like a box. ## \\ ## Edit: I think I see what they are wanting, but I'll let you work on it for a few minutes.
 
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  • #7
Charles Link said:
I am a little puzzled by this one also, because it doesn't look like a box.
I´ve translated it from my own language, so box is probably the wrong description. Let's say container instead. (edited first post)
 
  • #8
Oh, and you're also integrating under the curve, i.e. outside the box. Replace 0.5x2 by (4.5 - 0.5x2).
Try drawing a diagram of this to see what the "box" looks like (it's like a bowl).
 
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  • #9
This is what I think you're being asked to find the volume of. What is the volume of the bowl?
bowl.png
 

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  • #10
mjc123 said:
Oh, and you're also integrating under the curve, i.e. outside the box. Replace 0.5x2 by (4.5 - 0.5x2).
Try drawing a diagram of this to see what the "box" looks like (it's like a bowl).
Is it like this? We want to find the volume which results when rotating the gray colored area around x = -3, but we also need to include the "inner" volume which I´ve colored orange. The volume of this orange part should be pi*3^2*4.5..?

42982625_431839237221407_8407967947338809344_n.jpg
 

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  • #11
Looks like you've got it.
 
  • #12
mjc123 said:
Looks like you've got it.
I still get wrong. My answer is (135/2)*pi. Is it the same as you?
 
  • #13
No, I get 114.75*pi, by both methods.
 
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  • #14
mjc123 said:
No, I get 114.75*pi, by both methods.
Excellent, had forgotten that the graph was 0.5x^2 and not x^2... got it now!
 

What is the "Volume of Revolution"?

The Volume of Revolution is a mathematical concept that calculates the volume of a solid object formed by rotating a two-dimensional shape around an axis. This concept is commonly used in calculus and physics to calculate the volume of various objects such as cylinders, cones, and spheres.

What is the Shell Method?

The Shell Method is a specific technique used to calculate the volume of revolution. It involves dividing the object into thin cylindrical shells and then integrating them to find the total volume. This method is especially useful for objects with holes or irregular shapes.

What is the formula for finding the Volume of Revolution using the Shell Method?

The formula for the Volume of Revolution using the Shell Method is V = 2π∫(x)(f(x))dx, where x represents the distance from the axis of revolution and f(x) represents the function that defines the shape being rotated. This formula is derived from the volume of a cylinder (V = πr^2h) and takes into account the varying radius of the cylindrical shells.

What are the steps for finding the Volume of Revolution using the Shell Method?

The steps for finding the Volume of Revolution using the Shell Method are as follows:

  1. Identify the axis of rotation and the shape being rotated.
  2. Sketch the shape and the axis of rotation on a graph.
  3. Divide the shape into thin cylindrical shells perpendicular to the axis of rotation.
  4. Write an expression for the radius of each shell in terms of x.
  5. Write an expression for the height of each shell in terms of x.
  6. Use the formula V = 2π∫(x)(f(x))dx to calculate the volume of each shell.
  7. Add up the volumes of all the shells to find the total volume of the object.

What are some real-life applications of the Volume of Revolution and the Shell Method?

The Volume of Revolution and the Shell Method have various real-life applications, such as calculating the volume of objects in engineering, architecture, and physics. For example, the Shell Method can be used to calculate the volume of a water tank, a curved roof, or a rotating machine component. It is also commonly used in designing and analyzing 3D-printed objects.

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