Volume of Solid: Find Revolving Around x=-1

In summary, to find the volume of the solid generated by revolving about x = -1, we use the method of cylindrical shells and integrate the surface area of the thin cylinders formed by subtracting y = 0 from y = -x^2 + 4x -3, with a radius of x+1 and a thickness of ds. This differs from problems revolving around x = 0, where the radius is simply x and the thickness is also ds.
  • #1
science.girl
103
0
Problem:
Find the volume of the solid generated by revolving about the line x = -1, the region bounded by the curves y = -x^2 +4x -3 and y =0.

Attempt at a Solution:
I know that subtracting y = 0 from y = -x^2 +4x -3 will give the area in 1 dimension. So, would you use shells? I'm not sure how to set this up.

Additionally, the solid is being rotated around x = -1, so it must be different than problems with solids revolving around x = 0, correct?
 
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  • #2
science.girl said:
Problem:
Find the volume of the solid generated by revolving about the line x = -1, the region bounded by the curves y = -x^2 +4x -3 and y =0.

Attempt at a Solution:
I know that subtracting y = 0 from y = -x^2 +4x -3 will give the area in 1 dimension. So, would you use shells? I'm not sure how to set this up.
Since you are rotating around around x= -1 and are writing the formula in terms of x, yes, use "shells". "Subtracting y= 0 from y= -x^2+ 4x- 3" does NOT give area (its integral does). It give the height of the very thin cylinder forming the shell. What is the surface area? The volume of the thin cylinder is its surface area time its thickness, ds. Integrate theat.

Additionally, the solid is being rotated around x = -1, so it must be different than problems with solids revolving around x = 0, correct?
Only in that the radius of the cylindrical shells is x-(-1)= x+1, not x- 0= x.
 

1. What is the formula for finding the volume of a solid revolving around x=-1?

The formula for finding the volume of a solid revolving around x=-1 is V = π∫abf(x)2 dx, where a and b are the limits of the region being revolved and f(x) is the function representing the shape of the cross-section.

2. How do you determine the limits of integration for finding the volume of a solid revolving around x=-1?

The limits of integration for finding the volume of a solid revolving around x=-1 are determined by the points of intersection between the function representing the shape of the cross-section and the line x=-1. These points will serve as the limits of integration in the formula V = π∫abf(x)2 dx.

3. What is the difference between finding the volume of a solid revolving around x=-1 and y=-1?

The difference between finding the volume of a solid revolving around x=-1 and y=-1 is the axis of revolution. When revolving around x=-1, the axis of revolution is a vertical line, and the cross-sections will be perpendicular to this line. When revolving around y=-1, the axis of revolution is a horizontal line, and the cross-sections will be parallel to this line.

4. Can the volume of a solid revolving around x=-1 be negative?

No, the volume of a solid revolving around x=-1 cannot be negative. Volume is a measure of space, and space cannot have a negative value. If the calculated volume is negative, it means there was a mistake in the calculations.

5. Are there any special cases to consider when finding the volume of a solid revolving around x=-1?

Yes, there are two special cases to consider when finding the volume of a solid revolving around x=-1. The first is when the function representing the shape of the cross-section is negative. In this case, the calculated volume will be negative and must be multiplied by -1 to get the actual volume. The second case is when the region being revolved is between the line x=-1 and the x-axis. In this case, the limits of integration will be the x-values of the points of intersection with the x-axis.

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