Volume of solid revolving about y-axis

In summary, to find the volume of the solid generated by revolving the region bounded by the graph of y = x^3 and the line y = x between x = 0 and x = 1 about the y-axis, we need to integrate with respect to y using the formula \pi\int_{0}^{1}[(x^3)^2 - (x)^2]dy. Don't forget to include the Pi in the equation and to use the correct limits of integration.
  • #1
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Homework Statement


Find the volume of the solid generated by revolving the region bounded by the graph of
y = x3 and the line y = x,
between x = 0 and x = 1,
about the y-axis.


Homework Equations



[tex]\pi[/tex][tex]\overline{1}[/tex][tex]\int[/tex][tex]\underline{0}[/tex][R(x)[tex]^{2}[/tex]-[r(x)][tex]^{2}[/tex]dx

The Attempt at a Solution


x^6 - x^2 dx = x^7/7 - x^3/3 is where I get stuck.
 
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  • #2
Since you're revolving it around the y axis, you would probably want to integrate the areas with respect to y. Once you do that and find your new limits of integration, there shouldn't be much of a problem getting the answer.

Edit: And don't forget about that Pi in the equation, when you integrated with respect to x you omitted it.
 
Last edited:
  • #3
yes you want each slice perpendicular to the line you are rotating to. in your case, each slice will be (deltaY) high so you would integrate in terms of y, not x.

then it just becomes the integral of pi(Routside)^2-pi(Rinside)^2dy
 

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

The formula for finding the volume of a solid revolving about the y-axis is ∫(πr^2)dy, where r is the distance from the y-axis to the edge of the solid.

2. How do you determine the limits of integration for finding the volume of a solid revolving about the y-axis?

The limits of integration for finding the volume of a solid revolving about the y-axis are the y-values that correspond to the beginning and end of the solid. These can be determined by setting up the integral with respect to y and solving for the y-values.

3. Can the volume of a solid revolving about the y-axis be negative?

No, the volume of a solid revolving about the y-axis cannot be negative. Volume is a measure of space and cannot have a negative value.

4. How is the volume of a solid revolving about the y-axis different from the volume of a solid revolving about the x-axis?

The main difference is in the formula used to calculate the volume. The volume of a solid revolving about the y-axis uses the variable y and the distance from the y-axis to the edge of the solid, while the volume of a solid revolving about the x-axis uses the variable x and the distance from the x-axis to the edge of the solid.

5. Can the volume of a solid revolving about the y-axis be calculated using a different shape besides a cylinder?

Yes, the formula for finding the volume of a solid revolving about the y-axis can be used for any shape that can be formed by revolving a curve around the y-axis. This includes cones, spheres, and other irregular shapes.

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