Final answer:
The integral of a function that is continuous on a half-open interval [a, b) exists even if the function is discontinuous at b. The correct answer is B: ∫[a, b) f(x) dx exists and is the area under the curve from a to just before b.
Step-by-step explanation:
If f is continuous on the interval [a, b) and is discontinuous at b, then the integral ∫[a, b) f(x) dx refers to the total area under the curve of f(x) from a to just before b. Because f(x) is discontinuous at b, the integral does not include the value at b. However, the existence of such an integral is guaranteed by the fact that f is continuous on the half-open interval [a, b).
The correct answer to this question is B. ∫[a, b) f(x) dx exists and is defined as the area under the curve of f(x) from a to just before b. When considering the evaluation of integrals for continuous functions up to a point of discontinuity, one can use improper integrals to potentially assign a value to the area under the curve up to that point. In this case, though the function is discontinuous at b, the integral from a to b (non-inclusive) can still be calculated and carry a meaningful value, representing the sum of the areas of the infinitesimal strips from a right up to, but not including b.