First of all thank you for taking the time and effort for explaining my question. The method you showed works for certain kind of integral. Say we have

Now:

if we plug in

and

into equation (1) and solve for

we get:

Then plug in

and

into equation (3) and solve for

we get:

Again if we plug in

and

into equation (4) and solve for

we get:

So the upper and lower bound of

is

and

respectively and upper and

lower bound for

is

and

respectively.

But the answer is:

Why is the upper limit for

is

instead of

(what i found) and lower bound 0 instead of

(what i found)? Why didn't the method you showed didn't work here? Did i make a mistake here?

Note: I edited this post couple of times to be correct. Sorry about that.