6.4 The Fundamental Theorem of Calculus and Accumulation Functions

A definite integral $\int_a^b f(x),dx$ gives one number: the net accumulation over a fixed interval. Let the upper endpoint move and the integral becomes a function. The Fundamental Theorem of Calculus says how fast that function changes, and in doing so it links integration to differentiation.

Accumulation functions

An accumulation function records how much change has built up from a fixed starting point. Given a rate function $f$, define

$$ A(x)=\int_a^x f(t),dt. $$

Here, $a$ is fixed while $x$ varies. The variable $t$ is a dummy variable: it identifies the input inside the integral, while $x$ identifies the current endpoint and therefore the input of the new function $A$.

For example, let $q(t)$ be the flow rate of water into a tank, in gallons per hour, and define

$$ V(t)=200+\int_0^t q(s),ds. $$

The tank begins with $200$ gallons. The integral gives the net volume added after $t$ hours, so $V(t)$ gives the tank’s total volume. Notice the structural relationship:

$$ V'(t)=q(t). $$

The derivative of the accumulated total is the current rate of accumulation.

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus expresses the central relationship between integration and differentiation. In its accumulation-function form, if $f$ is continuous on an interval containing $a$, then

$$ A(x)=\int_a^x f(t),dt \quad\Longrightarrow\quad A'(x)=f(x). $$

This is the conceptual bridge between the two major operations of calculus:

$$ \text{rate of change} ;\xrightarrow{\text{integration}}; \text{accumulated change} $$

and

$$ \text{accumulated change} ;\xrightarrow{\text{differentiation}}; \text{rate of change}. $$

The second practical form evaluates a definite integral using an antiderivative. If $F'(x)=f(x)$, then

$$ \int_a^b f(x),dx=F(b)-F(a). $$

For the tank model, if $q(t)=3t^2+2$ gallons per hour, then an antiderivative is

$$ Q(t)=t^3+2t. $$

The amount of water added from $t=1$ to $t=4$ is

$$ \int_1^4(3t^2+2),dt

\left[t^3+2t\right]_1^4

(64+8)-(1+2)

$$

Thus, $69$ gallons enter the tank during that interval.

The first FTC application differentiates an accumulation function; the second evaluates an accumulated quantity using an antiderivative.

Misconception check

Misconception: “The integral always gives total amount.” A definite integral gives net accumulated change. If a rate becomes negative, its contribution reduces the result. Also, the bounds matter: $\int_a^b f(x),dx$ and $\int_b^a f(x),dx$ have opposite signs.

A second common error is writing an accumulation function as $\int_a^x f(x),dx$. The same symbol cannot clearly serve as both the moving endpoint and the integration variable. Write $\int_a^x f(t),dt$ instead.

AP Calculus alignment and retrieval check

Topics 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation and 6.4 The Fundamental Theorem of Calculus and Accumulation Functions are core AP Calculus AB and BC expectations within Unit 6: Integration and Accumulation of Change, weighted approximately 17–20% of either exam. They chiefly develop Skill 2.C: Identify a re-expression of mathematical information presented in a given representation and Skill 1.D: Identify an appropriate mathematical rule or procedure based on the relationship between concepts. They also require Mathematical Practice 1—Implementing Mathematical Processes, Mathematical Practice 2—Connecting Representations, Mathematical Practice 3—Justification, and Mathematical Practice 4—Communication and Notation.

Retrieval check: If $B(x)=\int_2^x (u^3-1),du$, identify $B'(x)$ and explain what $\int_2^5(u^3-1),du$ represents. The answers are

$$ B'(x)=x^3-1 $$

and the net accumulation of the rate $u^3-1$ from $u=2$ to $u=5$.