Changing the limits on Integrals

In summary, the conversation discusses the confusion around when to change the limits on a definite integral when using u-substitution. The speaker provides an example of an integral where the limits are not changed and another where they are changed. They also mention that the second integral is improper due to a discontinuity in the integrand. Finally, it is clarified that when making a substitution, the new limits should be u(a) and u(b) instead of a and b.
  • #1
kLPantera
43
0
I'm confused as to when to change the limits on a definite integral.

Ex. Integral with the limits a=1, b=5, 3/(x+1)dx

I set u = x+1 and du = dx

I used u-substitution and everything worked out fine.

However for this one...

Ex. Integral with the limits a = 0, b = 2, 6x^2/sqrt((x^3)-1)

I used u-substitution u = sqrt((x^3)-1) and so 2du = (6x^2)dx
However the book says I need to change the limits of the integral.

So I'm not sure when to change the limits of an integral. Can anyone help? Thanks =D
 
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  • #2
For the first integral, did you change the integration limits to [2, 6]?

For the second one, of course you must change the limits of the integral when you make a substitution, but also note that this is an improper integral since the integrand is discontinuous somewhere on [0, 2].
 
  • #3
For the first integral no I did not.
 
  • #4
Hi kLPantera! :smile:

You're integrating with respect to x from a to b.
When you substitute u=u(x) you change the expression to read "du" instead of "dx".
This means that exactly from this moment on you're integrating with respect to u.
The limits for u=u(x) are then u(a) and u(b).
 
  • #5
Ah I understand it now lol
 

1. What does it mean to change the limits on an integral?

Changing the limits on an integral refers to the process of adjusting the boundaries within which the integral is being evaluated. This can be done by either shifting the limits to a different interval or by converting the limits to a different form, such as from Cartesian coordinates to polar coordinates.

2. Why would someone want to change the limits on an integral?

There are several reasons why someone might want to change the limits on an integral. One common reason is to simplify the integral or make it easier to solve. Another reason might be to change the coordinate system in order to better match the problem at hand.

3. How do you change the limits on an integral?

The process for changing the limits on an integral depends on the specific problem and the desired outcome. In general, you can change the limits by using a substitution or transformation, or by using a different coordinate system. It is important to carefully consider the problem at hand and choose the most appropriate method for changing the limits.

4. What are the benefits of changing the limits on an integral?

As mentioned earlier, changing the limits on an integral can help simplify the problem or make it easier to solve. It can also provide a better understanding of the problem or make it easier to visualize. Additionally, changing the limits can sometimes lead to unexpected or interesting results that may not have been apparent with the original limits.

5. Are there any limitations or restrictions when changing the limits on an integral?

Yes, there are some limitations and restrictions when changing the limits on an integral. For example, when using a substitution or transformation, the new limits must still cover the same range of values as the original limits. Additionally, the new limits must still be valid for the function being integrated. It is important to carefully consider these limitations when changing the limits on an integral to ensure the accuracy and validity of the solution.

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