Integrable function

Definition

An integrable function is a function for which the definite integral over a specified interval exists and is finite. This concept ensures that the area under the curve of the function can be measured.

5 Must Know Facts For Your Next Test

1. A function must be bounded and continuous on a closed interval $[a, b]$ to be Riemann integrable.
2. If a function has a finite number of discontinuities on $[a, b]$, it may still be integrable.
3. The Fundamental Theorem of Calculus links the concept of an integrable function to antiderivatives.
4. Piecewise continuous functions are often integrable over their intervals of continuity.
5. Improper integrals extend the idea of integration to unbounded intervals or unbounded functions.

Review Questions

• What conditions must a function satisfy to be considered Riemann integrable?
• How does the Fundamental Theorem of Calculus relate to integrable functions?
• Can a function with discontinuities still be integrable? Under what conditions?

Related terms

Definite Integral: The definite integral of a function over an interval $[a, b]$ gives the net area under the curve from $a$ to $b$.

Riemann Sum: A method for approximating the total area under a curve by dividing it into smaller subintervals and summing up areas of rectangles formed within these subintervals.

Fundamental Theorem of Calculus:

$\text{Part I}$: It states that if $f$ is continuous on $[a, b]$, then its indefinite integral is differentiable and its derivative is $f$. \n$\text{Part II}$: It states that if $F$ is an antiderivative of $f$ on $[a, b]$, then \(\int_{a}^{b} f(x) \, dx = F(b) - F(a)\).