Fundamental Theorem of Calculus Cheat Sheet

A printable reference covering FTC Part 1, FTC Part 2, accumulation functions, net change, and average value for grades 11-12.

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The Fundamental Theorem of Calculus connects derivatives and definite integrals, showing that accumulation and rate of change are inverse ideas. This cheat sheet helps students recognize when to differentiate an integral, evaluate a definite integral using an antiderivative, or interpret accumulated change. It is especially useful for AP Calculus and precalculus-to-calculus review because it links graphs, formulas, and units. The main ideas are FTC Part 1, FTC Part 2, net change, and accumulation functions. FTC Part 1 says that if $F(x)=\int_a^x f(t)\,dt$ , then $F'(x)=f(x)$ when $f$ is continuous. FTC Part 2 says that $\int_a^b f(x)\,dx=F(b)-F(a)$ when $F'(x)=f(x)$ . These formulas explain why integrals measure accumulated change and derivatives measure instantaneous change.

Key Facts

FTC Part 1 states that if $F(x)=\int_a^x f(t)\,dt$ and $f$ is continuous, then $F'(x)=f(x)$ .
FTC Part 2 states that if $F'(x)=f(x)$ , then $\int_a^b f(x)\,dx=F(b)-F(a)$ .
For a variable upper limit, $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt=f(g(x))g'(x)$ by the chain rule.
For a variable lower limit, $\frac{d}{dx}\int_{g(x)}^a f(t)\,dt=-f(g(x))g'(x)$ .
Net change is calculated by $\int_a^b r(t)\,dt=Q(b)-Q(a)$ when $r(t)=Q'(t)$ .
Total change uses accumulated distance or amount and may require $\int_a^b |v(t)|\,dt$ instead of $\int_a^b v(t)\,dt$ .
The average value of a continuous function on $[a,b]$ is $f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx$ .
A definite integral can be negative when the graph of $f(x)$ lies below the $x$ -axis, because signed area is counted.

Vocabulary

Definite integral: A definite integral $\int_a^b f(x)\,dx$ represents the signed accumulation of $f(x)$ from $x=a$ to $x=b$ .
Antiderivative: An antiderivative of $f(x)$ is a function $F(x)$ such that $F'(x)=f(x)$ .
Accumulation function: An accumulation function has the form $A(x)=\int_a^x f(t)\,dt$ and gives the accumulated signed area up to $x$ .
Net change: Net change is the final amount minus the initial amount, written as $Q(b)-Q(a)=\int_a^b Q'(t)\,dt$ .
Signed area: Signed area counts regions above the horizontal axis as positive and regions below the horizontal axis as negative.
Average value: The average value of $f(x)$ on $[a,b]$ is $\frac{1}{b-a}\int_a^b f(x)\,dx$ .

Common Mistakes to Avoid

Forgetting the chain rule with variable limits is wrong because $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt$ equals $f(g(x))g'(x)$ , not just $f(g(x))$ .
Treating all definite integrals as positive area is wrong because $\int_a^b f(x)\,dx$ measures signed area, so parts below the axis subtract.
Using the integrand instead of an antiderivative in FTC Part 2 is wrong because $\int_a^b f(x)\,dx$ requires $F(b)-F(a)$ where $F'(x)=f(x)$ .
Confusing net change with total distance is wrong because $\int_a^b v(t)\,dt$ gives displacement, while $\int_a^b |v(t)|\,dt$ gives total distance.
Dropping the negative sign for a variable lower limit is wrong because $\frac{d}{dx}\int_{g(x)}^a f(t)\,dt=-f(g(x))g'(x)$ .

Practice Questions

1 Evaluate $\int_1^3 2x\,dx$ using the Fundamental Theorem of Calculus.
2 If $A(x)=\int_2^x (t^2+1)\,dt$ , find $A'(x)$ and $A'(3)$ .
3 Find $\frac{d}{dx}\int_0^{x^2} \sqrt{1+t^3}\,dt$ .
4 A velocity graph is sometimes above and sometimes below the time axis on $[0,6]$ . Explain why $\int_0^6 v(t)\,dt$ and $\int_0^6 |v(t)|\,dt$ can represent different physical quantities.

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