For a function to be integrable in a specific interval, it means that the function can be represented by a finite area under the curve within that interval. In other words, the function has a definite integral in that interval.
To determine if a function is integrable in a given interval, you can use the Riemann integral. This involves dividing the interval into smaller subintervals and calculating the area under the curve for each subinterval. If the sum of these areas approaches a finite value as the subintervals get smaller and smaller, then the function is considered integrable in that interval.
Yes, a function can be integrable in one interval but not another. This is because the Riemann integral only determines the integrability of a function within a specific interval. It is possible for a function to have a finite integral in one interval, but not in another.
The interval [-2,2] is significant because it is the specific interval in which we are trying to determine if the function is integrable. It is important to specify a specific interval when discussing integrability, as a function may have a finite integral in one interval but not in another.
Yes, a function can be continuous but not integrable in an interval. Being continuous is only one of the requirements for a function to be integrable. Other factors, such as the continuity of the derivative, also play a role in determining integrability.