Volume of a Solid: Solve Problem with Washer/Shell Method

In summary, the conversation was about finding the volume of the solid obtained by rotating a region bounded by given curves about a specified line using the washer and shell methods. The person had trouble with their attempt and asked for help. After some discussion and corrections, the correct answer was determined to be 424pi/3.
  • #1
willson.slp
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I'm have trouble with a homework problem having to do with find volume of an area. the problem reads:

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

y=x^2 , y=0 , x=2 , x=4;


about the line x=7

I've tried using the washer method and also the shell method with no luck. Any help would be greatly appreciated.

Thanks
 
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  • #2
Show your attempt, please, so as we can find out what went wrong.

ehild
 
  • #3
since its rotating about the line x=7, I solved for y first

Im using pi*intergral from 0 to 16 of ((7-sqrt(y))^2-(7-5)^2)dy

I think I'm messing up with what my inner and outer radius' are and possibly the bounds
 
Last edited:
  • #4
Y goes from 0 to x^2. First I would get the volume of a cylindrical shell of radius 7-x, thickness dx and height y=x^2. Then the integration with respect to x goes from x=2 to x=4.

ehild
 
Last edited:
  • #5
That did it!
I came up with 424pi/3

When I initially tried the shell method I was using just x for the radius instead of 7-x. Thanks again for the help!
 
  • #6
Is not it 436pi/3?

ehild
 
  • #7
I double checked it and still got 424pi/3
The homework is online and I got it correct
 

Related to Volume of a Solid: Solve Problem with Washer/Shell Method

1. What is the washer method and how is it used to find the volume of a solid?

The washer method is a mathematical technique used to calculate the volume of a solid that is formed by rotating a 2-dimensional region around a specific axis. This method involves slicing the solid into thin washers, calculating the volume of each washer using the formula π(router2 - rinner2)h, and then summing up the volumes of all the washers to find the total volume of the solid.

2. What is the shell method and when is it used to find the volume of a solid?

The shell method is another mathematical technique used to find the volume of a solid formed by rotating a 2-dimensional region around a specific axis. This method involves slicing the solid into thin cylindrical shells, calculating the volume of each shell using the formula 2πrh, and then summing up the volumes of all the shells to find the total volume of the solid. The shell method is typically used when the shape of the solid cannot be easily divided into washers or when the integration is simpler using cylindrical coordinates.

3. How do the washer and shell methods differ from each other?

The main difference between the washer and shell methods is the shape of the slices used to calculate the volume of the solid. The washer method uses circular washers, while the shell method uses cylindrical shells. Additionally, the washer method is used when the axis of rotation is perpendicular to the base of the solid, while the shell method is used when the axis of rotation is parallel to the base of the solid.

4. What are the steps involved in using the washer or shell method to find the volume of a solid?

The steps for using the washer or shell method to find the volume of a solid are as follows:

  • Step 1: Identify the axis of rotation and the 2-dimensional region that will be rotated to form the solid.
  • Step 2: Sketch the solid and the slices that will be used to calculate its volume.
  • Step 3: Determine the radius of the outer and inner edges of the slice (for washer method) or the radius and height of the shell (for shell method).
  • Step 4: Set up the integral by expressing the volume of each slice as a function of the variable of integration (typically x or y).
  • Step 5: Integrate the function over the appropriate interval to find the total volume of the solid.

5. Can the washer and shell methods be used interchangeably to find the volume of any solid?

No, the washer and shell methods are not interchangeable and can only be used for specific types of solids. The washer method is used for solids with a perpendicular axis of rotation, while the shell method is used for solids with a parallel axis of rotation. It is important to carefully identify the axis of rotation and choose the appropriate method when solving for the volume of a solid.

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