Fundamental Theorem of Calculus

Bridging Derivatives and Integrals

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The Fundamental Theorem of Calculus links two big ideas in calculus: accumulation and rate of change. Definite integrals measure accumulated quantity such as area, distance, or total change, while derivatives measure instantaneous change or slope. This theorem matters because it shows these two processes are inverse operations. It turns many hard area problems into easier function evaluations.

The theorem has two main parts. Part 1 says that if $F(x) = \int_a^x f(t)\,dt$ , then $F'(x) = f(x)$ , so the derivative of an accumulation function gives back the original function. Part 2 says that if $F$ is any antiderivative of $f$ , then $\int_a^b f(x)\,dx = F(b) - F(a)$ . Together, these results explain why shaded area, changing upper limits, and tangent slopes can all be represented in one connected picture.

Key Facts

FTC Part 1: If $F(x) = \int_a^x f(t)\,dt$ , then $F'(x) = f(x)$ .
FTC Part 2: If $F'(x) = f(x)$ , then $\int_a^b f(x)\,dx = F(b) - F(a)$ .
An antiderivative of $f$ is any function $F$ such that $F' = f$ .
The variable inside the integral is a dummy variable, so the name does not affect the value.
A definite integral gives signed area, so regions below the $x$ -axis contribute negative value.
Average value of $f$ on $[a, b]$ is $\frac{1}{b - a}$ times $\int_a^b f(x)\,dx$ .

Vocabulary

Definite integral: A definite integral gives the net accumulated value of a function over an interval.
Antiderivative: An antiderivative of f is a function whose derivative is f.
Accumulation function: An accumulation function is formed by integrating from a fixed starting point to a variable endpoint.
Dummy variable: A dummy variable is the integration variable inside an $\int$ and its name does not affect the value.
Signed area: Signed area counts area above the x-axis as positive and area below the x-axis as negative.

Common Mistakes to Avoid

Confusing the upper limit x with the integration variable, which is wrong because the inside variable is just a placeholder and should be different from the outside variable.
Forgetting that the definite integral gives signed area, which is wrong because parts of the graph below the $x$ -axis subtract from the total instead of adding.
Using $F(a) - F(b)$ instead of $F(b) - F(a)$ , which is wrong because the Fundamental Theorem uses upper endpoint minus lower endpoint in that order.
Adding $+ C$ to a definite integral answer, which is wrong because definite integrals evaluate to a number, while $+ C$ is only used for indefinite integrals.

Practice Questions

1 Let $F(x) = \int_1^x (3t^2 - 4)\,dt$ . Find $F'(x)$ and then find $F(2)$ .
2 Evaluate $\int_0^3 (2x + 5)\,dx$ using an antiderivative.
3 A function $f(x)$ is positive and increasing on $[a, b]$ . Explain what the Fundamental Theorem says about the slope of $G(x) = \int_a^x f(t)\,dt$ .