Final answer:
To decide between the First and Second Fundamental Theorem of Calculus, one must consider the problem context: the First Theorem relates derivatives to integrals and is used for finding accumulation, while the Second Theorem is for evaluating definite integrals when the antiderivative is known.
Step-by-step explanation:
To determine which fundamental theorem of calculus should be used, we must first understand what each one states. The First Fundamental Theorem of Calculus establishes the relationship between the derivative and the integral; that is, if a function f is continuous on an interval [a, b] and F is the indefinite integral of f on [a, b], then the derivative of F is f. This theorem allows us to evaluate the integral of a function over a range [a, b].
On the other hand, the Second Fundamental Theorem of Calculus provides an efficient way to calculate the definite integral of a function, particularly if we have its antiderivative. This theorem states that if F is an antiderivative of a continuous function f over an interval [a, b], then the integral of f from a to b is equal to F(b) - F(a).
As part of providing a solution strategy or using a calculator like the TI-83, 83+, or 84, these theorems play different roles. When we want to find the accumulation of a rate of change, we use the First Fundamental Theorem. If we already have the antiderivative and need to calculate the area under a curve between two points, we use the Second Fundamental Theorem.