Paradox? An infinite set having finite volume?

In summary, the conversation discusses finding the volume of a solid generated by rotating a region bounded by e^{-x} and the coordinate axes about the y-axis. The integral for this volume is not trivial to solve, but it can be evaluated using integration by parts. The answer is finite, but requires taking an appropriate limit.
  • #1
mrdoe
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0
Paradox? An infinite set having finite volume??

Homework Statement


Find the volume, in terms of k, of the solid made from R rotating about the y-axis, if R is defined as the region bounded by [tex]e^{-x}[/tex] and the coordinate axes.
2. The attempt at a solution
Obviously, [tex]\displaystyle\int^{1}_{0} (\ln (y))^2\pi dx = [/tex] undefined... but the region R itself has a finite volume. So if you rotate it shouldn't it have a finite volume?
 
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  • #2


I assume that you mean
[tex]
\displaystyle\int^{1}_{0} (\ln (y))^2\pi dy = [/tex]
because you are integrating over y.

The integral can be done, but it is not trivial to find the primitive.

My suggestion would be to "try"
[tex]y \ln^2(y)[/tex]
which upon differentiation gives you in any case the integrand back... then try to cancel out the unwanted term by adding something.

As for the evaluation of the integral, it looks like it is diverging. However, if you take an appropriate limit, you will still get the correct (finite) answer. The trick here is that your logs will only occur in terms like
[tex]y \ln y, y \ln^2 y, \cdots[/tex]
and that the linear term goes to zero faster than the logarithm goes to infinity, as y -> 0.
 
  • #3


You don't really have to guess the primitive. The systematic way to do the integral is to do integration by parts twice. Start with u=ln(y)^2 dv=dy.
 
  • #4


Yeah I'm getting the following for indefinite integral...

[tex]\pi x\ln^2 x - 2\pi x\ln x + 2\pi x[/tex]

But then if you try that with 0 it's undefined so... would the answer be [tex]2\pi[/tex]?
 
  • #5


CompuChip already said this, but I'll repeat. Things like x*ln(x) may be undefined at x=0, but they do have a limit as x->0. You do have to check that limit is 0 before you can ignore those terms. You can't ignore something just because it's 'undefined'.
 

1. What is a paradox?

A paradox is a statement or situation that seems to contradict itself or go against common sense, but may actually be true.

2. How can an infinite set have finite volume?

This paradox is known as "Hilbert's paradox of the Grand Hotel." It suggests that a hotel with an infinite number of rooms can still accommodate an infinite number of new guests, even if all of its rooms are occupied. This is because infinity is not a number, but a concept that represents endlessness.

3. What are some other examples of paradoxes?

Some other famous paradoxes include the liar paradox, which is a statement that says "this statement is false," and the grandfather paradox, which deals with time travel and the potential for changing the past.

4. How is this paradox relevant to science?

Paradoxes are relevant to science because they challenge our understanding of the world and push us to think critically and creatively. They can also lead to new discoveries and breakthroughs in scientific understanding.

5. How do scientists approach paradoxes in their research?

Scientists often use paradoxes as a starting point for their research, as they can reveal inconsistencies and gaps in current theories. They also try to resolve paradoxes by proposing new hypotheses and conducting experiments to test them.

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