Fundamental Theorem of Calculus

Summary

The Fundamental Theorem of Calculus (FTC) links differentiation and integration. Part 1 says that the integral-from-a-fixed-lower-limit defines an antiderivative. Part 2 evaluates definite integrals using any antiderivative.

Prerequisites

Limits, Derivatives, Integrals, Riemann Sum

Definition / Theorem

Let $f$ be continuous on the closed interval $[a,b]$ .

Part 1

Define

Then $F$ is differentiable on $(a,b)$ and $F^{\prime }(x)=f(x)$ . (At the endpoints one uses one-sided derivatives.)

Part 2

If $F$ is any antiderivative of $f$ on $[a,b]$ (so $F^{\prime }=f$ ), then

Conditions / Assumptions

• Continuity of $f$ on $[a,b]$ is a standard sufficient hypothesis for both parts as stated in elementary calculus.
• Part 2 still holds under weaker integrability conditions if $F^{\prime }=f$ almost everywhere and $F$ is absolutely continuous, but that is beyond this note.
• The lower limit in Part 1 may be any fixed point in the interval; changing it only shifts $F$ by a constant.

Worked Example

1. If $G(x)={\int }_0^x{t}^{2}dt$ , then $G^{\prime }(x)=x^2$ by Part 1.
2. ${\int }_1^3(2x+1)dx$ : an antiderivative is $F(x)=x^2+x$ , so

Common Mistakes

• Applying Part 2 without an antiderivative that is valid on the whole interval.
• Forgetting the chain rule when the upper limit is a function: $\frac{d}{dx}{\int }_a^{u(x)}f(t)dt=f(u(x))u^{\prime }(x)$ .

Connections

• Builds on Riemann Sum and Derivatives
• Used constantly in Integrals and applications

References

Both parts appear with continuity hypotheses in OpenStax Calculus Volume 1.^[1]