Final answer:
To prove that a function is integrable, it must satisfy Riemann's Criterion for integrability, which states that the function should be bounded and have a set of discontinuities with measure zero. By applying Riemann's Criterion to the given functions, we can prove their integrability.
Step-by-step explanation:
To prove that a function is integrable, we need to show that the function satisfies Riemann's Criterion for integrability. According to Riemann's Criterion, if a function is bounded and its set of discontinuities has measure zero (meaning the set of points where the function is not continuous has no length, area, or volume), then the function is integrable. Let's apply Riemann's Criterion to the two functions given:
(i) For the function f(x) = 3x² + 5x + 9 defined on the interval (0,10], we can see that it is a polynomial function and therefore continuous everywhere. Polynomials are bounded on any finite interval, so f(x) is bounded on (0,10]. Additionally, since f(x) is a polynomial, it is continuous everywhere, and therefore it has no points of discontinuity. Hence, the set of discontinuities has a measure of zero. Therefore, f(x) satisfies Riemann's Criterion and is integrable on (0,10].
(ii) For the function g(x) = [x] defined on the interval [1,100], we can observe that g(x) is the greatest integer function, which is also known as the floor function. The greatest integer function is discontinuous at every integer. However, on the interval [1,100], the only integers are 1, 2, 3, ..., 100. Since the set of integers is countable, its measure is zero. Therefore, the set of discontinuities of g(x) has measure zero. Hence, g(x) satisfies Riemann's Criterion and is integrable on [1,100].