Integrability of Continuous and Monotonic Functions A function is Riemann integrable on a closed interval [a, b] if the Riemann integral of that fun...
Integrability of Continuous and Monotonic Functions
A function is Riemann integrable on a closed interval [a, b] if the Riemann integral of that function exists. The Riemann integral is an infinite sum of the areas of rectangles under the curve.
Important Points:
A function is monotonic on [a, b] if it is either increasing or decreasing.
A function is continuous on [a, b] if its graph has no breaks or jumps within that interval.
To calculate the Riemann integral, we break the interval [a, b] into a finite number of subintervals and add the areas of the rectangles.
The Riemann integral exists if the limit of the sum of the areas of the rectangles as the number of rectangles approaches infinity is finite.
Darboux Sums
A function is Darboux integrable on a closed interval [a, b] if the Darboux sum of that function exists. The Darboux sum is a finite sum of the function evaluated at specific points in the interval.
Comparison between Riemann Integrability and Darboux Integrability:
A function is Riemann integrable if it is both continuous and monotonic on the interval [a, b].
A function is Darboux integrable if it is only continuous on the interval [a, b].
Examples:
The function f(x) = x^2 is Riemann integrable on [0, 1] with the Riemann integral being 1/3.
The function f(x) = x is Darboux integrable on [0, 1] with the Darboux sum being the same as the Riemann integral.
The function f(x) = x^3 is Riemann integrable on [0, 1] but not Darboux integrable on [0, 1]