Improper integral concept help.

In summary, the conversation discusses an example of an improper integral and how it can be split into two separate integrals. It is determined that the second integral diverges, which means that the original integral also diverges. This means that there is no answer and the expression is meaningless, so evaluating the original improper integral should be stopped.
  • #1
frasifrasi
276
0
Ok, my book has the example int from -1 to 2 of dx/x^3

this gets split into int from -1 to 0 of dx/x^3 and int from 0 to 2 of dx/x^3.

Now, he had previously determined that the second integral returned positive infinity (diverges) by taking the lim as b approaches 0+.


So, the book goes on to say that since the second integral diverges the original integral also diverges.

--> but what does that mean? is "divergers" the answer to "evaluate the int from -1 to 2 of dx/x^3" ? Or does it mean the the answer is just infinity?

Can anyone explain?
 
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  • #2
It means that it diverges.
 
  • #3
so what is the answer ? what does this mean ?? what should I conclude from this? does it mean that I shoudl stope evaluating the original improper integral?

I would appreciate HELPFUL input. If you aren't going to clarify the solution, please igonre the thread.
 
  • #4
frasifrasi said:
so what is the answer ?
There is none.
what does this mean ?? what should I conclude from this?
That the expression is meaningless.
does it mean that I shoudl stope evaluating the original improper integral?
Quite so.
 

1. What is an improper integral?

An improper integral is a type of integral that does not have a finite value. This can occur when the limits of integration are infinite or when the integrand has an infinite value at a certain point within the limits.

2. What is the difference between a proper and an improper integral?

A proper integral has finite limits of integration and a continuous integrand within those limits, resulting in a finite value. An improper integral, on the other hand, may have infinite limits or a discontinuous integrand, resulting in an infinite or undefined value.

3. How do you evaluate an improper integral?

To evaluate an improper integral, the integral is first split into smaller integrals with finite limits. Then, the limit of these smaller integrals is taken as the limits of integration approach infinity or a point of discontinuity. If the limit exists, it represents the value of the improper integral.

4. What is the significance of improper integrals in real-world applications?

Improper integrals are useful in modeling real-world scenarios where values may approach infinity or have discontinuities. For example, they are commonly used in physics to calculate areas under curves that represent the motion of objects with changing velocities.

5. Can improper integrals be solved using numerical methods?

Yes, improper integrals can be solved using numerical methods such as the trapezoidal rule or Simpson's rule. These methods approximate the value of the integral by dividing the area under the curve into smaller trapezoids or parabolas, respectively, and summing their areas.

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