Fundamental Theorem of Calculus

Connecting Derivatives and Integrals

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The Fundamental Theorem of Calculus is one of the most important ideas in mathematics because it shows that differentiation and integration are inverse processes. Differentiation measures instantaneous change, while integration measures accumulated quantity such as area under a curve. This theorem lets students move between rates and totals in a precise way. It is the bridge that makes much of applied calculus possible in physics, engineering, economics, and biology.

The theorem has two closely connected parts. The first part says that if you build an accumulation function from a continuous function $f$ , then the derivative of that accumulation is the original function: $\frac{d}{dx} \int_a^x f(t)\,dt = f(x)$ . The second part says that a definite integral can be computed using any antiderivative $F$ of $f$ : $\int_a^b f(x)\,dx = F(b) - F(a)$ . Together these ideas connect slope and area, local behavior and total change.

Key Facts

If $F(x) = \int_a^x f(t)\,dt$ and $f$ is continuous, then $F'(x) = f(x)$ .
If $F'(x) = f(x)$ , then $\int_a^b f(x)\,dx = F(b) - F(a)$ .
Differentiation gives rate of change, while definite integration gives accumulated change.
The variable inside the integral, as in $\int_a^x f(t)\,dt$ , is a dummy variable and can be replaced by another symbol.
Net area can be negative when the graph of $f(x)$ lies below the $x$ -axis.
For motion, if $v(t)$ is velocity, then displacement on $[a, b]$ is $\int_a^b v(t)\,dt$ .

Vocabulary

Antiderivative: An antiderivative of $f$ is a function $F$ whose derivative is $f$ , so $F'(x) = f(x)$ .
Definite integral: A definite integral gives the accumulated net change of a function over an interval.
Accumulation function: An accumulation function measures total area or total change from a fixed starting point up to x.
Continuous function: A continuous function has no breaks, jumps, or holes on the interval being studied.
Net change: Net change is the final amount minus the initial amount, often computed with an $\int$ of a rate.

Common Mistakes to Avoid

Treating the definite $\int$ as always positive, which is wrong because areas below the $x$ -axis contribute negative signed area to the $\int$ .
Forgetting to evaluate the antiderivative at both bounds, which is wrong because $\int_a^b f(x)\,dx$ equals $F(b) - F(a)$ , not just $F(b)$ .
Using the same variable for the limit and the integration variable in an accumulation function, which is wrong because writing $\int_a^x f(x)\,dx$ confuses the changing upper limit with the dummy variable.
Assuming the theorem works without the needed conditions, which is wrong because continuity of f is required for the first part in the standard form and differentiability of F is required when using an antiderivative.

Practice Questions

1 Let $f(x) = 3x^2 + 2$ . Find $\int_1^4 f(x)\,dx$ .
2 Define $G(x) = \int_0^x (t^3 - 5t)\,dt$ . Find $G'(x)$ and then evaluate $G'(2)$ .
3 A function $f(x)$ is positive and increasing on the interval $[a, b]$ . Explain what the Fundamental Theorem of Calculus says about the derivative of $A(x) = \int_a^x f(t)\,dt$ and describe what that means about the graph of $A$ .

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