Finding volume using integration

In summary, the conversation is discussing the area of a solid when the area between two curves is rotated about a specific axis. The person has attempted to solve the problem and has attached their solution, but it does not match the answer key. They are questioning why their area formula (pi-pi*y^4) is wrong and wondering if it should be pi*(1-y^2)^2 instead. They are also asking for an explanation of why their formula may be incorrect.
  • #1
theBEAST
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Homework Statement


Given x=y^2, x=1 what is the area of the solid when the area between the two curves is rotated about x=1.

The Attempt at a Solution


I attached my solution and according to the answer key my area formula (pi-pi*y^4) is wrong. Instead they have pi*(1-y^2)^2. Can anyone explain why my area formula is wrong? I thought that it was the area of the upper boundary curve minus the area of the lower boundary curve...
 

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  • #2
theBEAST said:

Homework Statement


Given x=y^2, x=1 what is the area of the solid when the area between the two curves is rotated about x=1.

The Attempt at a Solution


I attached my solution and according to the answer key my area formula (pi-pi*y^4) is wrong. Instead they have pi*(1-y^2)^2. Can anyone explain why my area formula is wrong? I thought that it was the area of the upper boundary curve minus the area of the lower boundary curve...
What method are you trying to use? Is it disks? ... or is it shells?
 

1. What is the concept of volume in integration?

Volume in integration refers to the calculation of the space occupied by a three-dimensional object. It involves breaking down the object into infinitesimally small pieces and finding the sum of their volumes using mathematical integration.

2. How is the volume of a solid determined using integration?

To determine the volume of a solid using integration, the solid is divided into smaller and simpler shapes such as cubes, cylinders, or spheres. The volume of each shape is then calculated using the appropriate formula, and these volumes are then summed up using integration to find the total volume of the solid.

3. What is the difference between finding volume using integration and using traditional methods?

The traditional method of finding volume involves measuring the dimensions of the solid and using formulas such as length x width x height or base area x height. Integration, on the other hand, involves using mathematical calculations to find the sum of infinitesimally small volumes, allowing for more accurate and precise results.

4. What are some real-life applications of finding volume using integration?

Finding volume using integration has many practical applications in fields such as architecture, engineering, and physics. It is used to calculate the volume of irregularly shaped objects, such as buildings, bridges, and tunnels. It is also used in fluid dynamics to determine the volume of liquids or gases in a given space.

5. What are some tips for successfully finding volume using integration?

Some tips for successfully finding volume using integration include carefully choosing the limits of integration, understanding the shape and dimensions of the solid, and using the appropriate integration formula for each shape. It is also crucial to double-check calculations and be mindful of units to ensure accurate results.

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