Mean Value Theorem to calculate solids of revolution?

In summary, the conversation discusses using the Mean Value Theorem to calculate the volume of a solid of revolution, specifically in the case of rotating y=e^x across the x-axis between x=2 and x=3. While the disk method is commonly used, it is suggested that the average height of the curve, found using the Mean Value Theorem, could also be used to calculate the volume. However, it is explained that this method falls short because the average value is not equivalent to the centroid, which is needed to use Pappus' theorem.
  • #1
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Mean Value Theorem to calculate solids of revolution?

Ive been studying calculus on my own because my school doesn't offer it and i came across solids of revolution tonight. In one of the problems it says "What is the volume of the solid formed by rotating y=e^x across the x-axis between x=2 and x=3 ?" They did it using the disk method. It occurred to me however, that if you used the mean value theorem to find the average height of the curve, that would give you the average radius, so then u should just be able to use (pi)(r^2)(height) to find the volume, but height is just 3-2=1 . But it always seems to come up short. Does anybody know why?
 
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  • #2
The height of the equation is e^x. I think that is the problem. Maybe not though I looked briefly.
 
  • #3
The "average value" of the height of the graph is NOT the "centroid" and that's what you would need to use. Look up "Pappus' theorem" in your textbook:
 

1. What is the Mean Value Theorem?

The Mean Value Theorem is a mathematical theorem that states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point within the interval where the slope of the tangent line is equal to the average rate of change of the function.

2. How is the Mean Value Theorem used to calculate solids of revolution?

The Mean Value Theorem can be used to find the volume of a solid of revolution by setting up an integral that represents the cross-sectional area of the solid. The theorem can then be applied to find the radius of a circle that has the same area as the cross-section. This radius can be used to calculate the volume of the solid.

3. What are the assumptions of the Mean Value Theorem?

The Mean Value Theorem assumes that the function is continuous on a closed interval and differentiable on the open interval. It also assumes that the function is not constant, meaning that it has different values at different points within the interval.

4. Can the Mean Value Theorem be used for all types of functions?

No, the Mean Value Theorem can only be used for functions that satisfy the assumptions of the theorem. In particular, it cannot be used for discontinuous or non-differentiable functions.

5. How is the Mean Value Theorem related to the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is a generalization of the Mean Value Theorem. It states that if a function is continuous on a closed interval and has a continuous derivative, then the definite integral of the derivative of the function over the interval is equal to the difference of the function evaluated at the endpoints of the interval. This can be seen as a special case of the Mean Value Theorem, where the average rate of change is equal to the derivative of the function.

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