Find the volume of the solid of revolution.

In summary, the volume of the solid of revolution obtained by rotating the area bounded by the curves y = x^2 – 2 and y = 0 about the line y = -1 is calculated by taking the distance between the point (x,y) and the line y = -1, which is 1 - |y|, and integrating it over the given limits. This results in the equation 1 - |y| = 1 - (-y) = 1 + y = y + 1. Finally, the distance of the point (x,y) from the x-axis is subtracted from this equation to obtain the final answer.
  • #1
s3a
818
8

Homework Statement


Problem:
Find the volume of the solid of revolution obtained by rotating the area bounded by the curves y = x^2 – 2 and y = 0 about the line y = -1. Consider only that part above y = -1.

Solution:
The solution is attached as TheSolution.jpeg.

Homework Equations


Integration.

The Attempt at a Solution


I get everything the solution did except for the part with the PQ = 1 - |y| = 1 - (-y) = 1 + y = y + 1.

Could someone please explain that part for me?

Any input would be greatly appreciated!
 

Attachments

  • TheSolution.jpg
    TheSolution.jpg
    42.8 KB · Views: 393
Physics news on Phys.org
  • #2
It is just the radius of revolution of the dydx element, : ##y_{upper}-y_{lower}=y -(-1)##.
 
  • #3
Sorry, I double-posted.
 
  • #4
Y_lower = the line about which we are rotating, right?

As for the upper part of the radius, I don't see why y_upper = y.
 
  • #5
s3a said:
Y_lower = the line about which we are rotating, right?

As for the upper part of the radius, I don't see why y_upper = y.

You have a little dydx square located at the variable point (x,y) in the interior of your area that you are integrating over the area. It is the distance from that variable (x,y) point to the line y = -1 that is the radius of revolution.
 
  • #6
Why is ##y_upper## = y (a variable) if the parabola is bounded above by a constant function (y = 0)?

Why isn't it ##y_upper – y_lower## = 0 - (-y) = y (such that the third dimension of the volume is 2π × y instead of 2π × (y + 1))?

Also, why does the solution say 1 - |y| = 1 - (-y) = 1 + y = y + 1 instead of y - (-1) = y + 1? (I know the final answer is the same; I'm asking about the difference in getting to the final answer.)
 
  • #7
s3a said:
Why is ##y_{upper}## = y (a variable) if the parabola is bounded above by a constant function (y = 0)?

You want the radius of rotation of the little dydx square. You want its distance from ##y=-1##.

Why isn't it ##y_{upper} – y_{lower}## = 0 - (-y) = y (such that the third dimension of the volume is 2π × y instead of 2π × (y + 1))?

We are talking about the upper and lower ends of the radius of rotation. The upper end is at the dydx square and the lower end is ##y=-1##.

Also, why does the solution say 1 - |y| = 1 - (-y) = 1 + y = y + 1 instead of y - (-1) = y + 1? (I know the final answer is the same; I'm asking about the difference in getting to the final answer.)

I have no idea why he wrote it that way.
 
  • #8
s3a said:
Also, why does the solution say 1 - |y| = 1 - (-y) = 1 + y = y + 1 instead of y - (-1) = y + 1? (I know the final answer is the same; I'm asking about the difference in getting to the final answer.)
I'd guess that the way the person who wrote the solution was thinking about it is that the distance from the line y=-1 to the point (x,y) is 1, the distance between the x-axis and the line y=-1, minus |y|, the distance from (x,y) to the x-axis.
 
  • #9
Thanks guys, I get it now! :)
 

What is the solid of revolution?

The solid of revolution is a three-dimensional object created by rotating a two-dimensional curve or shape around an axis.

Why is it important to find the volume of the solid of revolution?

Calculating the volume of the solid of revolution is important in many areas of science, engineering, and mathematics. It is used to solve real-world problems such as finding the volume of a water tank, the amount of material needed to create a specific shape, or the capacity of a container.

What is the formula for finding the volume of the solid of revolution?

The formula for finding the volume of the solid of revolution is V = π∫a^b y^2 dx, where a and b represent the limits of integration, y is the function of the curve being rotated, and dx is the differential of x.

What are the steps for finding the volume of the solid of revolution?

To find the volume of the solid of revolution, follow these steps:

1. Identify the axis of rotation and the limits of integration.

2. Determine the function of the curve being rotated.

3. Use the formula V = π∫a^b y^2 dx to set up the integral.

4. Evaluate the integral to find the volume.

What are some common examples of finding the volume of the solid of revolution?

Some common examples of finding the volume of the solid of revolution include:

- Finding the volume of a sphere by rotating a semicircle around its diameter.

- Calculating the volume of a cone by rotating a right triangle around one of its legs.

- Determining the volume of a wine bottle by rotating a parabola around its axis of symmetry.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
903
  • Calculus and Beyond Homework Help
Replies
1
Views
854
  • Calculus and Beyond Homework Help
Replies
1
Views
883
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
Replies
6
Views
944
  • Calculus and Beyond Homework Help
Replies
13
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
905
  • Calculus and Beyond Homework Help
Replies
5
Views
660
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
940
Back
Top