Calculating Volume of Overlapping Regions using Integration

In summary, if you are working with a region that overlaps, you must determine the limits where the overlap takes place in order to calculate the proper area.
  • #1
daivinhtran
68
0

Homework Statement


THe region bounded by y = -x + 3 and y = x^2 - 3x
the region revolve about a, x-axis, and b, y=axis

Homework Equations


V = π∫r^2 dx

The Attempt at a Solution


I have no clue to solve it since the volume overlap. I try to ignore the overlapped region but didn't get the right answer.
 
Last edited:
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  • #2
Your problem statement as worded will not result in a volume, but an area.

Have you left something out, like the region is to be rotated about a certain axis?

In any event, you must determine the limits where the overlap takes place, in order to calculate the proper area.
 
  • #3
SteamKing said:
Your problem statement as worded will not result in a volume, but an area.

Have you left something out, like the region is to be rotated about a certain axis?

In any event, you must determine the limits where the overlap takes place, in order to calculate the proper area.

oh yes...I actually left something out...the region revolve about a, x-axis, and b, y=axis..
 
  • #4
Treat the overlapping and non-overlapping regions separately.
 
  • #5
what do you do with the overlapping region?
 
  • #6
daivinhtran said:
what do you do with the overlapping region?
The following is a graph by WolframAlpha which may help for the rotation about the x axis.

attachment.php?attachmentid=55357&stc=1&d=1359950154.gif


It's a graph of 4 curves:

[itex]y = -x + 3[/itex]

[itex]y = -(-x + 3)[/itex]

[itex]y = x^2 - 3x[/itex]

[itex]y = -(x^2 - 3x)[/itex]
 

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  • #7
SammyS said:
The following is a graph by WolframAlpha which may help for the rotation about the x axis.

attachment.php?attachmentid=55357&stc=1&d=1359950154.gif


It's a graph of 4 curves:

[itex]y = -x + 3[/itex]

[itex]y = -(-x + 3)[/itex]

[itex]y = x^2 - 3x[/itex]

[itex]y = -(x^2 - 3x)[/itex]

I did try this way...I find the region at the interval [-1, 0] by take the integrate of pi ∫(-x+3)^2 - (x^2 - 3x)^2 dx...
Then, pi x ∫ (-x+3)^2 - (-x^2 + 3x)^2 dx for interval [0,1]
Then, pi x ∫ (3x-x^2)^2 for the interval [1,3]
and take the sum of all...and get 56pi/3

but the answer is not that
 
  • #8
daivinhtran said:
...

Then, pi x ∫ (-x+3)^2 - (-x^2 + 3x)^2 dx for interval [0,1]
...

This one is incorrect.

Added in Edit:

For x > 0, the inner radius is zero. -- This becomes the disc method rather than the washer method for x > 0.
 
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  • #9
I'll modify my previous suggestion. If you treat overlapping and non-overlapping entirely separately you'll have five separate integrals (2 for overlap, 3 for non).
It's a bit easier to process the whole shape without worrying about the overlap, then subtract the overlap parts on the basis that they have been counted twice. That should reduce it to the three integration steps.
 

What is the concept of finding volumes by integration?

Finding volumes by integration is a mathematical technique used to calculate the volume of a three-dimensional shape by breaking it down into infinitesimally small slices and adding them together using integration.

What are the steps involved in finding volumes by integration?

The first step is to determine the limits of integration, which define the boundaries of the shape. Then, the shape is divided into small slices, and the volume of each slice is calculated using the appropriate integration method. Finally, the volumes of all the slices are added together to get the total volume of the shape.

What types of shapes can be solved using integration to find their volumes?

Integration can be used to find the volume of any three-dimensional shape, including simple shapes like cubes and spheres, as well as more complex shapes like cones, cylinders, and toroids.

What is the difference between using integration and other methods, such as geometric formulas, to find volumes?

Integration offers a more accurate and versatile method for finding volumes, as it can be applied to any shape, regardless of its complexity. Geometric formulas, on the other hand, are limited to specific shapes and may not provide an accurate result for more complex shapes.

What are the real-life applications of finding volumes by integration?

Finding volumes by integration has numerous applications in fields such as physics, engineering, and architecture. It can be used to calculate the volume of objects, such as containers and buildings, and to solve real-world problems, such as determining the amount of fluid in a tank or the strength of a bridge.

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