Is Pi Finite? A Calculus Question

In summary, the conversation discusses the concept of volume of a solid of revolution formed by rotating the curve y=1/x around the x-axis. The integral for finding the volume is V=\pi\int^{z}_{1}1/x^2 dx, which when evaluated for large values of z, converges to a finite value of \pi. Although \pi is an irrational number, this does not mean that it is infinite, as it still has a defined value and does not continuously increase as x tends to infinity.
  • #1
tpingt
2
0
I'm relatively new to calculus, and this question was bugging me, so I have decided to ask it.
We have the function [tex]y=1/x[/tex] with domain [tex]x\geq1[/tex] and we rotate the curve around the x-axis in order to form a solid of revolution. (Gabriel's Horn)
The integral is [tex]V=\pi\int^{z}_{1}1/x^2 dx[/tex] and we evaluate to [tex]V=\pi(1-1/z)[/tex]
Take the limit as z approaches infinity: [tex]\lim_{z \to \infty}\pi(1-1/z)=\pi[/tex]

Apparently my teacher says that the volume is a finite amount, which is to say [tex]\pi[/tex]. However isn't pi an irrational number? Meaning that its digits keep going without end? If pi is not finite, then how can the volume be finite? Wouldn't the volume keep getting minutely closer to pi as x tends to infinity?

Thanks!
 
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  • #2
What he means when he says it is "finite" is that it converges to a real value (such as [itex]\pi[/itex]), instead of diverging to [itex]\infty[/itex].
 
  • #3
Welcome to PF!

Hi tpingt! Welcome to PF! :smile:
tpingt said:
… Take the limit as z approaches infinity: [tex]\lim_{z \to \infty}\pi(1-1/z)=\pi[/tex]

Apparently my teacher says that the volume is a finite amount, which is to say [tex]\pi[/tex]. However isn't pi an irrational number? Meaning that its digits keep going without end? If pi is not finite, then how can the volume be finite? Wouldn't the volume keep getting minutely closer to pi as x tends to infinity?

Yes, π is irrational, and so its digits keep going without end and without repetition …

but that doesn't make it infinite …

it doesn't even make it more than 4 …

or more than 3.2 …

or more than 3.15 …

or … well, you get the idea. :smile:
 
  • #4
Thanks for the great explanations guys, I understand it now! :)
 

1. Is Pi a finite number?

No, Pi (π) is an irrational number, meaning it has an infinite number of digits after the decimal point and cannot be expressed as a simple fraction.

2. Can Pi be calculated using calculus?

Yes, Pi can be calculated using calculus by evaluating the integral of the function 1/(1+x^2) from 0 to 1. This is known as the Leibniz formula for Pi.

3. How is Pi related to calculus?

Pi is related to calculus through the study of geometric shapes, specifically circles. The ratio of a circle's circumference to its diameter is always equal to Pi, and calculus can be used to find the area and volume of a circle.

4. Why is Pi considered an important number in mathematics?

Pi is considered an important number in mathematics because it is a fundamental constant that appears in many mathematical equations and formulas. It is also used in real-world applications such as engineering, physics, and statistics.

5. Is there an end to the digits of Pi?

No, there is no end to the digits of Pi. It has been calculated to over 31 trillion digits and there is no known pattern to the digits. It is believed to be a truly random and infinite number.

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