Finding the Volume of a Solid by Revolving a Region

In summary, to find the volume of the solid generated by revolving the region bounded by y = x^2 - 4x + 5 and y = 5 - x about the line y = -1, you need to find the points of intersection and use the appropriate integral formula. The given integral is for the area between the two curves and not for the volume. You need to consider the shape of a thin slice rotated around the line y = -1 to determine the correct integral for finding the volume.
  • #1
science.girl
103
0
Problem:
Find the volume of a solid generated by revolving the region bounded by the graphs of y = x^2 - 4x +5 and y = 5- x about the line y = -1.

Attempt:
Do I find where the graphs intersect, then make those values the upper and lower bounds for the integral?

And could I set the problem up by having the integral of: [((5 - x) -1) - ((x^2 - 4x +5) - 1)]dx



Thanks! (And my apologies -- I'm having issues with the template.)
 
Physics news on Phys.org
  • #2
science.girl said:
Problem:
Find the volume of a solid generated by revolving the region bounded by the graphs of y = x^2 - 4x +5 and y = 5- x about the line y = -1.

Attempt:
Do I find where the graphs intersect, then make those values the upper and lower bounds for the integral?

And could I set the problem up by having the integral of: [((5 - x) -1) - ((x^2 - 4x +5) - 1)]dx



Thanks! (And my apologies -- I'm having issues with the template.)
Yes, you need to find the points where they intersect. But that integral looks like the integral for the area between the the two curves, not the volume of the rotated region. Imagine a thin slice between, say, x and dx, rotated around the line y= -1. What figure does that look like? What is its volume?
 

1. What does it mean to "revolve a region" to find the volume of a solid?

When we talk about revolving a region, we mean rotating a two-dimensional shape around an axis to create a three-dimensional solid. This is often done by taking a cross-sectional area and rotating it around a vertical or horizontal axis.

2. How do you find the volume of a solid by revolving a region?

To find the volume of a solid by revolving a region, we use a mathematical technique called integration. This involves breaking the solid down into thin slices, calculating the volume of each slice, and adding them together to get the total volume.

3. What is the difference between revolving a region around a horizontal axis and a vertical axis?

Revolving a region around a horizontal axis creates a solid with a circular cross-section, while revolving a region around a vertical axis creates a solid with a cross-section in the shape of a washer (a ring with a hole in the middle). The method for finding the volume is the same in both cases, but the formula for the cross-sectional area will be different.

4. Can you use this method to find the volume of any solid?

No, this method can only be used for finding the volume of solids that have a circular or circular-like cross-section. If the solid has a different shape, such as a triangle or rectangle, other methods will need to be used to find the volume.

5. Are there any limitations to using this method for finding volume?

One limitation is that the shape being revolved must have a known equation or function that describes it. Additionally, the limits of integration must be known and defined in order to calculate the volume accurately. If these requirements are not met, other methods for finding volume may need to be used.

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
905
  • Calculus and Beyond Homework Help
Replies
8
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
883
  • Calculus and Beyond Homework Help
Replies
27
Views
2K
  • Calculus and Beyond Homework Help
Replies
10
Views
204
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
402
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
864
  • Calculus and Beyond Homework Help
Replies
3
Views
904
Back
Top