How does this second integral equal +infinity instead of -infinity?

In summary: The first is the limit of (\frac{-1}{a - 1} - \frac{-1}{0 - 1}) as a approaches 1 from below, the second is the limit of (\frac{-1}{2 - 1} - \frac{-1}{b - 1}) as b approaches 1 from above, and the third is the limit of (\frac{-1}{c - 1} - \frac{-1}{2 - 1}) as c approaches infinity. The first is easy: as a approaches 1 from below, \frac{-1}{a - 1} approaches -\infty and \frac{-1}{0 - 1}= -1 so the limit is
  • #1
Lo.Lee.Ta.
217
0
1. ...How does this 2nd integral diverge to +∞? It seems to me that it would diverge to -∞... :/

lim ([itex]\frac{-1}{a - 1}[/itex] - [itex]\frac{-1}{0 - 1}[/itex]) + lim ([itex]\frac{-1}{2 - 1}[/itex] - [itex]\frac{-1}{b - 1}[/itex]) + lim ([itex]\frac{-1}{c - 1}[/itex] - [itex]\frac{-1}{2 - 1}[/itex])
a→1- b→1+ c→∞

2. First integral: -1/(tiny negative #) - 1 = +∞

Second integral: -1 + 1/(tiny negative #) = -1 + -∞ = -∞ <----This answer is supposed to be +∞. How?!

Third Integral: -1/(∞ - 1) + 1 = 0 + 1 = 0

The only thing I don't get it the second integral! :(
Help please!
Thanks! :)
 
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  • #2
Lo.Lee.Ta. said:
1. ...How does this 2nd integral diverge to +∞? It seems to me that it would diverge to -∞... :/

lim ([itex]\frac{-1}{a - 1}[/itex] - [itex]\frac{-1}{0 - 1}[/itex] + lim ([itex]\frac{-1}{2 - 1}[/itex] - [itex]\frac{-1}{b-1}[/itex]
a→1- b→1+
+ lim ([itex]\frac{-1}{c - 1}[/itex] - [itex]\frac{-1}{2 - 1}[/itex]
c→∞


2. First integral: -1/(tiny negative #) - 1 = +∞

Second integral: -1 + 1/(tiny negative #) = -1 + -∞ = -∞ <----This answer is supposed to be +∞. How?!
It is NOT "tiny negative". [itex]-\frac{-1}{b- 1}= \frac{1}{b- 1}[/itex] and b is approaching 1 from above so b> 1 and b- 1> 0.

Third Integral: -1/(∞ - 1) + 1 = 0 + 1 = 0

The only thing I don't get it the second integral! :(
Help please!
Thanks! :)
 
  • #3
If b is approaching 1 from the right, it seems like it would be very close to the number 1.

Maybe it would be 1.00000000001

So 1/(1.00000000001 + 1) = 1/.00000000001 = +∞

I usually pay no attention to the - or + in the limit, but I see it's important! O_O
Thanks!
 
  • #4
These are not integrals: they are limits.
 

FAQ: How does this second integral equal +infinity instead of -infinity?

How does the second integral equal +infinity instead of -infinity?

The value of a definite integral depends on the function being integrated and the limits of integration. In the case of the second integral, the function being integrated may have an asymptote or a singularity at the lower limit of integration, causing the integral to diverge to +infinity instead of -infinity.

Can a definite integral have a value of +infinity?

Yes, a definite integral can have a value of +infinity. This occurs when the function being integrated has a singularity or an asymptote at one or both of the limits of integration, causing the integral to diverge to +infinity.

What is the difference between a definite integral equaling +infinity and diverging?

When a definite integral equals +infinity, it means that the integral has a finite value of +infinity. This occurs when the function being integrated has a singularity or an asymptote at one or both of the limits of integration. On the other hand, when a definite integral diverges, it means that the integral does not have a finite value and continues to increase or decrease without bound.

Can the value of a definite integral change from +infinity to -infinity?

Yes, the value of a definite integral can change from +infinity to -infinity. This can happen when the function being integrated changes from having a singularity or an asymptote at the lower limit of integration to having one at the upper limit of integration, or vice versa. In this case, the integral will have a different value depending on the direction of integration.

How can I determine if a definite integral will equal +infinity or -infinity?

To determine the value of a definite integral, you need to evaluate the integral using appropriate techniques such as substitution or integration by parts. If the function being integrated has a singularity or an asymptote at one or both of the limits of integration, the integral may diverge to +infinity or -infinity. In general, it is important to carefully consider the function being integrated and the limits of integration to determine the value of a definite integral.

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