FTC and Definite Integrals for AP Calculus AB — AP Calculus AB
1. The Second Fundamental Theorem of Calculus (FTC 2): Evaluating Definite Integrals ★★☆☆☆ ⏱ 4 min
FTC 2 gives the method for computing exact values of definite integrals using antiderivatives, eliminating the need to compute limits of infinite Riemann sums. The theorem applies when $f(x)$ is continuous on the closed interval $[a,b]$, and $F(x)$ is any antiderivative of $f(x)$ (meaning $F'(x) = f(x)$ for all $x$ in $[a,b]$).
int_a^b f(x) dx = F(b) - F(a)
The standard notation for this evaluation is $\left. F(x) \right|_a^b$, which explicitly means 'evaluate $F$ at the upper bound $b$, subtract the value of $F$ at the lower bound $a$.' The constant of integration $+C$ can be omitted entirely for definite integrals, because it cancels out when subtracting $F(a)$ from $F(b)$. FTC 2 can only be used if $f$ is continuous over the entire interval of integration; discontinuities like vertical asymptotes invalidate direct application of the theorem.
Exam tip: Always write the evaluation as $\left. F(x) \right|_a^b$ explicitly to avoid mixing up the order of bounds. Reversing the bounds will flip the sign of your answer, which is a common avoidable error.
2. The First Fundamental Theorem of Calculus (FTC 1): Derivatives of Accumulation Functions ★★★☆☆ ⏱ 4 min
FTC 1 formalizes the inverse relationship between integration and differentiation, and is heavily tested on AP Calculus AB multiple-choice questions involving accumulation functions (functions defined as integrals). The base case for a constant lower bound and variable upper bound $x$ is:
\frac{d}{dx} \left[ \int_a^x f(t) dt \right] = f(x)
When the upper bound is a function of $x$ ($u(x)$) rather than just $x$, we apply the chain rule to get the general form. If both bounds are functions of $x$, we split the integral at a constant to get the extended rule. The extended rule for $g(x) = \int_{v(x)}^{u(x)} f(t) dt$ is:
g'(x) = f(u(x))u'(x) - f(v(x))v'(x)
Exam tip: When the variable of differentiation is in the lower bound, do not drop the required negative sign from splitting the integral. Double-check the sign of the lower bound term before submitting your final answer.
3. The Net Change Theorem: FTC for Contextual Problems ★★★☆☆ ⏱ 3 min
The Net Change Theorem is a direct application of FTC 2 to real-world problems, where we integrate a rate of change to find total accumulated change over an interval. It states that the net change in a quantity $F(t)$ from $t=a$ to $t=b$ is equal to the definite integral of its rate of change $f(t) = F'(t)$ over that interval:
\text{Net Change} = F(b) - F(a) = \int_a^b f(t) dt
This theorem is the foundation for all contextual integration problems on AP Calculus AB, including particle displacement, population change, drug amount in the bloodstream, and total profit from marginal profit. Net change counts positive and negative change against each other (e.g., moving left vs. right for a particle), unlike total distance which requires integrating the absolute value of the rate. If asked for the final amount of a quantity at $t=b$ given an initial amount $F(a)$, rearrange the formula to get: $F(b) = F(a) + \int_a^b f(t) dt$.
Exam tip: When asked for the final amount of a quantity, the definite integral only gives the net change, not the final amount. Always add the net change to the given initial amount to get the correct answer.
4. AP Style Concept Check ★★★☆☆ ⏱ 3 min
Common Pitfalls
Why: Students remember FTC gives the integrand evaluated at the upper bound, but forget the chain rule when the bound is a function of $x$.
Why: Students forget FTC 2 requires $f$ to be continuous over the entire interval of integration, and this function has a discontinuity at $x=0$ inside the interval.
Why: Students mix up the order of evaluation, especially when bounds are written in reverse order in the problem.
Why: Students confuse net change with total final amount.
Why: Students are used to the constant lower bound case and forget the negative sign is required when the variable is in the lower bound.