Integrals and the Fundamental Theorem — AP Calculus AB
1. Antiderivatives and Indefinite Integrals ★★☆☆☆ ⏱ 3 min
Core integration rules are direct inverses of common differentiation rules:
- Power rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$
- Trigonometric rules: $\int \sin x dx = -\cos x + C$, $\int \cos x dx = \sin x + C$
- Constant multiple rule: $\int kf(x) dx = k\int f(x) dx$
- Sum/difference rule: $\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$
Exam tip: You will lose 1 point per missing $+C$ on indefinite integral free response questions, so write it immediately after computing the antiderivative to avoid forgetting.
2. Riemann Sum Approximations ★★☆☆☆ ⏱ 4 min
A Riemann sum estimates the area under a curve $f(x)$ over $[a,b]$, which is the definition of the definite integral $\int_a^b f(x) dx$. To calculate a Riemann sum, divide the interval into equal-width subintervals, then sum the area of rectangles with heights from $f(x)$ at a sample point in each subinterval.
For $n$ equal subintervals, the width of each subinterval is:
\Delta x = \frac{b-a}{n}
Three common types of Riemann sums are defined by which sample point is used:
- **Left Riemann sum**: Uses the left endpoint of each subinterval for height: $L_n = \Delta x \sum_{i=0}^{n-1} f(x_i)$
- **Right Riemann sum**: Uses the right endpoint of each subinterval for height: $R_n = \Delta x \sum_{i=1}^{n} f(x_i)$
- **Midpoint Riemann sum**: Uses the midpoint of each subinterval for height: $M_n = \Delta x \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\right)$
3. Fundamental Theorem of Calculus (Parts 1 & 2) ★★★☆☆ ⏱ 4 min
The Fundamental Theorem of Calculus (FTC) unifies differentiation and integration, allowing exact calculation of definite integrals without approximation. It has two complementary parts, both heavily tested on the AP exam.
4. u-Substitution Method ★★★☆☆ ⏱ 4 min
u-substitution is the inverse of the chain rule for differentiation, used to integrate composite functions of the form $f(g(x)) \cdot g'(x)$. The step-by-step process is:
- Identify the inner composite function $g(x)$ whose derivative (or a constant multiple of its derivative) is present in the integrand
- Set $u = g(x)$, compute $du = g'(x) dx$, and rearrange to solve for $dx$ if needed
- Rewrite the entire integral in terms of $u$, eliminating all $x$ terms
- Integrate with respect to $u$
- For indefinite integrals: substitute $g(x)$ back in for $u$ and add $+C$
- For definite integrals: calculate new $u$-bounds using $u = g(x)$ at $x=a$ and $x=b$, then evaluate at the new bounds (no need to substitute back to $x$)
5. Average Value of a Continuous Function ★★☆☆☆ ⏱ 2 min
Exam tip: Do not confuse average value with average rate of change, which is $\frac{f(b)-f(a)}{b-a}$. Average value describes the average height of the function, while average rate of change describes the average slope of the function.
Common Pitfalls
Why: Students focus on computing the antiderivative and skip $C$, since derivatives eliminate constants
Why: Students memorize sum formulas without mapping them to subinterval endpoints
Why: Students memorize FTC Part 1 only for integrals with a constant lower bound
Why: Students forget that $u$ is a different variable with different bounds
Why: Both formulas include the $\frac{1}{b-a}$ factor, leading to mix-ups