Answer
The definite integral exists when the function is Riemann integrable, ensuring that it meets certain conditions for boundedness and continuity over the closed interval \([a, b]\).
Work Step by Step
The definite integral of a function \( f \) over a closed interval \([a, b]\) is a number that represents the signed area between the graph of the function and the x-axis over that interval. It is denoted by the symbol \(\int_{a}^{b} f(x) \, dx\).
The definite integral exists when the function \( f \) is Riemann integrable over the interval \([a, b]\). Riemann integrability ensures that the function is bounded and has a finite number of discontinuities (of a certain type) within the interval. The function should not have infinite oscillations, and the size of these oscillations should approach zero as the width of the subintervals used in the Riemann sum approaches zero.
