Finding limits of integration during a change of variables

[Shinjo]

Hi. I have a problem with a question. Basically, I have an integral that goes from x=0 to x=1, and I'm supposed to make a change of variables like this:

Let x = 1 - y^2.

The problem I'm having is trying to find the limits of integration after the change of variables. Since y = +/- (x-1)^1/2, I have y = -1 or +1 for the lower limit. Am I supposed to just use either one? I'm so confused.

 

[arildno]

You need a bijection between the x's and the y's in this case.
That is, you should choose
EITHER $$y=\sqrt{1-x}$$
OR: $$y=-\sqrt{1-x}$$
Either one of them is okay; using both is wrong.

 

[M98Ranger]

Hi there,

Thank you for reaching out for help with your problem. It can definitely be tricky to find the limits of integration when making a change of variables, but don't worry, it is a common struggle for many students.

First, let's review the process of changing variables in an integral. When we substitute a new variable into an integral, we also need to adjust the limits of integration to reflect the new variable. In your case, you have substituted x=1-y^2, so you need to adjust the limits from x=0 to x=1 to y=? to y=?.

To find the new limits, we need to solve for y in terms of x. You have correctly found that y= +/- (x-1)^1/2, but we need to simplify this further. Remember, when we are changing variables, we are essentially changing the variable of integration, so we need to express the new variable in terms of x.

In this case, we can simplify y= +/- (x-1)^1/2 to just y= (x-1)^1/2. This is because we are only concerned with the positive values of y, since negative values of y will result in negative values of x, which are already covered by the original limits of integration.

So, the new limits of integration are y=0 to y=1, since y= (x-1)^1/2 and we are integrating with respect to y.

In summary, when making a change of variables, it is important to solve for the new variable in terms of the old variable and then adjust the limits of integration accordingly. I hope this helps clarify things for you. Good luck with your problem!