Units 6-8 cover integration — the reverse of differentiation. The Fundamental Theorem of Calculus connects derivatives and integrals, enabling calculation of areas, volumes, and solutions to differential equations.
FTC Part 1: d/dx[∫ₐˣ f(t)dt] = f(x). FTC Part 2: ∫ₐᵇ f(x)dx = F(b) - F(a) where F\' = f. Riemann sums: left, right, midpoint, trapezoidal — approximate area. u-substitution: reverse chain rule. Choose u = inside function, find du. Integration by parts (BC): ∫u dv = uv - ∫v du. Partial fractions (BC): decompose rational functions.
Area between curves: ∫ₐᵇ [f(x) - g(x)] dx (f on top). Volume by disc: V = π∫ₐᵇ [R(x)]² dx (rotation about axis, solid). Volume by washer: V = π∫ₐᵇ [R(x)² - r(x)²] dx (hollow). Volume by known cross-sections: V = ∫ₐᵇ A(x) dx where A(x) = area of cross-section. Shell method (BC): V = 2π∫ₐᵇ r(x)h(x) dx.
Separation of variables: dy/dx = f(x)g(y) → ∫(1/g(y))dy = ∫f(x)dx. Slope fields: graphical representation of dy/dx at each point. Euler\'s method (BC): numerical approximation. Logistic growth (BC): dP/dt = kP(1 - P/L), solution approaches carrying capacity L.
FTC bridges two seemingly different ideas: (1) derivatives (instantaneous rate of change) and (2) integrals (accumulated area/total). Part 1 says that if you define a function by accumulating area under f from a to x, then the derivative of that accumulation function is just f(x) — differentiation undoes integration. Part 2 says you can evaluate a definite integral by finding any antiderivative and evaluating at the endpoints — you don\'t need to sum infinitely many rectangles. Together, they make calculus practical: to find total accumulated quantity, just find an antiderivative.
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