Volume of a Solid Revolved About X-Axis

In summary, the conversation discusses finding the volume of a solid generated by revolving a region bounded by two equations around the x-axis. The process involves finding the points of intersection, using the washer method, and evaluating the integral π ∫ (from 0 to 1) of (x2/3 - x8) dx. It is noted that the two equations only intersect at zero and one, and the integral can be integrated from zero to one.
  • #1
Zach Hughes
2
0
Originally posted in a technical math section, so missing the template
I'm trying to practice for my final. The sample problem is:
"Find the volume of the solid generated when the region bounded by y = x4and y = x1/3, 0<=x<=1, is revolved about the x-axis."

To start, I set the two y equations equal to each to find the points of intersection.
x4 = x1/3, : raise both sides to power of 3
x12, = x
x - x12 = 0
So the intersection points that work are: x = 0, 1, -(-1)5/11, and (-1)6/11.

I believe I have now have to use the washer method to solve the problem.

With those multiple intersection points, it really confuses me on where to go next.

After looking through some examples in the text, I believe the integral I would have to evaluate to the find the correct volume is:
π ∫ (from 0 to 1) of (x2/3 - x8) dx

But I don't know how to get to that from the given information and the points of intersection I found. Any tips would be greatly appreciated. Thanks.
 
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  • #2
The two functions only intersect at zero and one right?

So just integrate the integral you have from zero to one.
 

1. What is the formula for finding the volume of a solid revolved about the x-axis?

The formula for finding the volume of a solid revolved about the x-axis is V = π∫(f(x))^2 dx, where f(x) is the function representing the solid's cross-sectional area at any given x-value.

2. Can the volume of a solid revolved about the x-axis be negative?

No, the volume of a solid revolved about the x-axis cannot be negative. It represents the amount of space occupied by the solid and therefore cannot have a negative value.

3. How do you determine the limits of integration for finding the volume of a solid revolved about the x-axis?

The limits of integration can be determined by finding the x-values where the solid intersects with the x-axis. These points will serve as the lower and upper limits of integration.

4. What is the difference between finding the volume of a solid revolved about the x-axis and the y-axis?

The main difference is in the formula used. When finding the volume of a solid revolved about the x-axis, the function representing the cross-sectional area is squared, while for finding the volume revolved about the y-axis, the function is cubed.

5. Can the volume of a solid revolved about the x-axis be calculated using calculus?

Yes, the volume of a solid revolved about the x-axis can be calculated using calculus by setting up an integral and evaluating it using the fundamental theorem of calculus. This is the most accurate method for finding the volume of a solid revolved about the x-axis.

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