## Summary of the Fundamental Theorem of Calculus

### Essential Concepts

• The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a value $c$ such that $f(c)$ 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 $f(x)$ is continuous over an interval $\left[a,b\right],$ then there is at least one point $c\in \left[a,b\right]$ such that $f(c)=\frac{1}{b-a}{\displaystyle\int }_{a}^{b}f(x)dx.$
• Fundamental Theorem of Calculus Part 1
If $f(x)$ is continuous over an interval $\left[a,b\right],$ and the function $F(x)$ is defined by $F(x)={\displaystyle\int }_{a}^{x}f(t)dt,$ then ${F}^{\prime }(x)=f(x).$
• Fundamental Theorem of Calculus Part 2
If $f$ is continuous over the interval $\left[a,b\right]$ and $F(x)$ is any antiderivative of $f(x),$ then ${\displaystyle\int }_{a}^{b}f(x)dx=F(b)-F(a).$

## 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 $c$ exists such that $f(c)$ is equal to the average value of the function