Solids of revolution - What if the axis crosses the volume?

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Homework Help Overview

The problem involves finding the volume of a solid generated by rotating the area bounded by the functions y=x^3 and y=4x around the line y=4. The original poster expresses confusion regarding the setup, particularly because the axis of rotation intersects the bounded area.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to set up the problem using integrals but encounters negative volume, leading to questions about the correct approach. Some participants question the points of intersection and suggest verifying the functions involved.

Discussion Status

Participants are exploring different interpretations of the problem, with some suggesting that the original function may have been misprinted. There is a discussion about how to proceed with the original problem despite the potential error, with one participant proposing a method involving the union of volumes generated by the areas above and below the line.

Contextual Notes

There is uncertainty regarding the correct functions and points of intersection, as well as the implications of the axis of rotation crossing the bounded area. The original poster has not provided a drawn representation of the problem, which may affect understanding.

Jani08
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Homework Statement



Here is the problem :

Find the volume of the solid generated when the area bounded between the following functions rotates along the y=4 line.

Functions: y=x^3 y=4x

Homework Equations



Functions: y=x^3 y=4x
Axis of rotation: y=4

The Attempt at a Solution



Points of intersection are x=-2,0,2
I tried making 2 integrals from -2 to 0 and 0 to 2, but I just ended up with a negative volume.
What I don't understand is that the axis of rotation crosses the area bounded these two lines make. How do I go about this? Never encountered thissituation. Maybe I am setting up the problem wrong...
 
Last edited:
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The curves do not intersect when x = -2. Show us your integral. Did you draw a picture?
 
Oh I am sorry it is y=x^3 not y=x^2, yeah I drew a picture but haven't scanned it.
 
If the function is in fact x3, you are correct that the axis of rotation passes through the region. I suspect a misprint in your text because of that. In fact, the problem would make perfect sense if the function was x2, which is certainly a likely candidate for what the problem was supposed to be, especially given the limits and the axis of rotation you have.
 
I agree with LCK, but what if we wanted to go on with the original problem ?
The volume would be the union of the two volumes separately generated by the two areas (above and below the line).
Then I'd mirror one area over the other taking the line as pivot, then make the union of the areas and proceed with rotation
 

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