The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a value [latex]c[/latex] such that [latex]f(c)[/latex] equals the average value of the function. See the Mean Value Theorem for Integrals.

The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral. See the Fundamental Theorem of Calculus, Part 1.

The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula. See the Fundamental Theorem of Calculus, Part 2.

Key Equations

Mean Value Theorem for Integrals
If [latex]f(x)[/latex] is continuous over an interval [latex]\left[a,b\right],[/latex] then there is at least one point [latex]c\in \left[a,b\right][/latex] such that [latex]f(c)=\frac{1}{b-a}{\displaystyle\int }_{a}^{b}f(x)dx.[/latex]

Fundamental Theorem of Calculus Part 1
If [latex]f(x)[/latex] is continuous over an interval [latex]\left[a,b\right],[/latex] and the function [latex]F(x)[/latex] is defined by [latex]F(x)={\displaystyle\int }_{a}^{x}f(t)dt,[/latex] then [latex]{F}^{\prime }(x)=f(x).[/latex]

Fundamental Theorem of Calculus Part 2
If [latex]f[/latex] is continuous over the interval [latex]\left[a,b\right][/latex] and [latex]F(x)[/latex] is any antiderivative of [latex]f(x),[/latex] then [latex]{\displaystyle\int }_{a}^{b}f(x)dx=F(b)-F(a).[/latex]

Glossary

fundamental theorem of calculus

the theorem, central to the entire development of calculus, that establishes the relationship between differentiation and integration

fundamental theorem of calculus, part 1

uses a definite integral to define an antiderivative of a function

fundamental theorem of calculus, part 2

(also, evaluation theorem) we can evaluate a definite integral by evaluating the antiderivative of the integrand at the endpoints of the interval and subtracting

mean value theorem for integrals

guarantees that a point [latex]c[/latex] exists such that [latex]f(c)[/latex] is equal to the average value of the function