Unit 6: Integration & Accumulation of Change
1. The Fundamental Theorem of Calculus (free preview)
The FTC connects the exam's two halves: derivatives and integrals are inverse operations. Two parts, both essential:
$\dfrac{d}{dx}\displaystyle\int_a^x f(t)\,dt = f(x) \qquad\qquad \displaystyle\int_a^b f(x)\,dx = F(b) - F(a)$
Part 1: differentiating an accumulation function hands back the integrand. Part 2: a definite integral is an antiderivative evaluated at the endpoints and subtracted. Between them they turn area problems into algebra.
Worked example
Evaluate $\displaystyle\int_1^3 2x\,dx$.
- Antiderivative of $2x$: $F(x) = x^2$.
- FTC part 2: $F(3) - F(1) = 9 - 1$.
- The integral is $8$.
Common mistake: when the upper limit is something like $x^2$ rather than plain $x$, FTC part 1 needs a chain rule factor. $\frac{d}{dx}\int_0^{x^2} \cos t\,dt = 2x\cos(x^2)$, not just $\cos(x^2)$.
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The rest of Unit 6 is in the full course
The largest question bank of any unit, matching its 17–20% exam weight:
- Every antiderivative rule: powers, trig, exponentials, $\tfrac{1}{x}$
- Riemann sums: left, right, midpoint, trapezoid, and over/under logic
- u-substitution, from first examples through definite integrals
- Accumulation functions and integral properties
- 50 exam-style questions with verified step-by-step solutions
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