Final answer:
The improper integral ∫(a to b) f(x) dx is called convergent if the corresponding limit exists and is finite. The term divergent is used when the limit does not exist or is infinite. The correct option for the described scenario is A) Convergent.
Step-by-step explanation:
The improper integral ∫(a to b) f(x) dx is an integral where either the limits of integration are infinite or the integrand, f(x), approaches infinity within the limits of integration. In order for this type of integral to be meaningful, we need to take a limit as we approach the boundary of integration. If the limit exists and is finite, we describe the integral as convergent. On the other hand, if the limit does not exist or is infinite, we call the integral divergent. Thus, an improper integral ∫(a to b) f(x) dx is called convergent if the corresponding limit exists and finite.
When evaluating an improper integral, we typically rewrite the integral with a limit expression. For example, if we are dealing with an integral from a to infinity, we would write ∫(a to ∞) f(x) dx as limit as t approaches infinity of ∫(a to t) f(x) dx. If we can find a finite value for this limit, the improper integral is convergent. If not, it is divergent. The convergence or divergence of an improper integral tells us about the behavior of the accumulated area under the curve of f(x) from x1 to x2 or beyond, represented in Figure 7.8.
Therefore, the correct option for the question is A) Convergent, since that is the term used for an improper integral that has a finite limit.