Calculating Volume of Solid Formed by Revolving Region

In summary: So, the volume of the region is pi*int with the boundaries from 0 to 1 being y=x^3 and the boundaries from 1 to 2 being y=1. So, the volume is pi*int(0 to 1) x^6 dx + pi*int(1 to 2)1 dx = pi(x^7/7)| from 0 to 1 + pi(x)| from 1 to 2 = (pi/7)*1 + pi = 8pi/7. In summary, the volume of the solid formed by revolving the region bounded by y=x^3, y=1, and x=2 around the x-axis is 8pi/7.
  • #1
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Homework Statement



Find the volume of a solid formed by revolving the region bounded by graphs of:
y=x^3
y=1
and
x=2


Homework Equations


[tex]\pi[/tex]0[tex]\int[/tex]2(x^3)dx


The Attempt at a Solution



x^7/7 with boundaries of [0,2]

Am I on the right path?
 
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  • #2
That's ok if you ignore the y=1 boundary, but I don't think you should. Draw a picture. I presume you are rotating around the x-axis?
 
  • #3
frumdogg said:

Homework Statement



Find the volume of a solid formed by revolving the region bounded by graphs of:
y=x^3
y=1
and
x=2
Revolved around what axis?


Homework Equations


[tex]\pi[/tex]0[tex]\int[/tex]2(x^3)dx
If you are the x-axis and are using the disk method, you would be integrating [itex]\pi\int_0^1 y^2 dx+ \pi\int_1^2 1 dx[/itex]


The Attempt at a Solution



x^7/7 with boundaries of [0,2]

Am I on the right path?
 
  • #4
Yes, rotating around the x-axis.
 
  • #5
frumdogg said:
Find the volume of a solid formed by revolving the region bounded by graphs of:
y=x^3
y=1
and
x=2

Hi frumdogg! :smile:

That doesn't look solid … do you mean y = 2 ? :confused:
 
  • #6
The problem says
y=x^3
y=1
x=2

When graphing it, it's a small area with x^3 on the left, y=1 on top, x=2 on right, and x-axis on bottom, at least what I am coming up with.
 
  • #7
hmm … they might as well have said:

y=x^3
x=1;

the rest is just a cylinder. :confused:
 
  • #8
frumdogg said:
The problem says
y=x^3
y=1
x=2

When graphing it, it's a small area with x^3 on the left, y=1 on top, x=2 on right, and x-axis on bottom, at least what I am coming up with.

It's a region with y=0 at the bottom and y=x^3 at the top from x equal 0 to 1 and y=1 at the top and y=0 at the bottom from x equal 1 to 2. Halls already set it up for you. tiny-tim is saying the same thing.
 

1. How do you calculate the volume of a solid formed by revolving a region?

The volume of a solid formed by revolving a region can be calculated using the formula V = π∫ba(f(x))2dx, where a and b represent the limits of the region and f(x) is the equation of the curve being revolved.

2. What is the difference between using the disk method and the shell method to calculate volume?

The disk method involves slicing the region into thin disks and adding up their volumes, while the shell method involves slicing the region into thin cylindrical shells and adding up their volumes. Both methods can be used to calculate volume, but the choice between them depends on the shape of the region being revolved.

3. How does the choice of axis affect the calculation of volume for a revolved region?

The choice of axis affects the calculation of volume because it determines the orientation of the slices used in the calculation. For example, if the axis is horizontal, the slices would be disks, while if the axis is vertical, the slices would be cylindrical shells.

4. Can the volume of a solid formed by revolving a region be negative?

No, the volume of a solid cannot be negative. If the calculation results in a negative value, it means that the region was incorrectly defined or the axis of rotation was incorrectly chosen.

5. What are some real-life applications of calculating volume of a solid formed by revolving a region?

The calculation of volume for a revolved region has many applications in fields such as engineering, architecture, and physics. For example, it can be used to calculate the volume of a 3D object created by rotating a 2D design, such as a vase or a bottle. It is also used in calculating the volume of rotational objects in physics, such as gears and turbines.

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