Unit 6: Integration and Accumulation of Change

Executive Summary

Unit 6 introduces the integral as accumulated change and the Fundamental Theorem connecting integration to differentiation — the largest unit on the exam (17–20%).

Riemann sums and the definite integral

$\int_a^b f(x)\,dx$ is the signed area between the curve and the x-axis. Left, right, midpoint, and trapezoidal sums approximate it; for monotone functions, left sums under- and right sums over-approximate (or vice versa for decreasing).

Antiderivatives

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ($n \neq -1$), $\int \frac{1}{x} dx = \ln|x| + C$, $\int e^x dx = e^x + C$, $\int \sin x\,dx = -\cos x + C$, $\int \cos x\,dx = \sin x + C$.

Fundamental Theorem of Calculus

Part 1: $\frac{d}{dx} \int_a^x f(t)\,dt = f(x)$ (chain: multiply by the inner derivative). Part 2: $\int_a^b f(x)\,dx = F(b) - F(a)$ for any antiderivative $F$.

u-substitution

Reverse chain rule: pick $u$, compute $du$, rewrite the integral entirely in $u$, and change bounds (or back-substitute). Long division and completing the square prep rational integrands.

Exam traps

$+C$ on every indefinite integral. Signed area can be negative — "area between curves" takes absolute values. FTC Part 1 requires the upper limit to be the variable — split or negate when bounds are reversed. Substitution must convert the ENTIRE integral, including $dx$.