A definite integral is accumulated change and signed area
The definite integral ∫₀ᵇ f(x) dx measures the signed area between f and the x-axis on [a, b], which equals the accumulated change of a quantity whose rate is f. You approximate this area with Riemann sums — left, right, and midpoint rectangles, or the trapezoidal rule — and as the number of subintervals grows without bound, the sum approaches the integral exactly: ∫₀ᵇ f dx = lim Σ f(xᵢ)Δx.
Definite IntegralRiemann SumsAccumulation
Key Concept 2
The Fundamental Theorem of Calculus connects derivatives and integrals
The FTC is the bridge of the course. Its evaluation form says ∫₀ᵇ f(x) dx = F(b) − F(a) for any antiderivative F of f — so integrals can be evaluated exactly once you can antidifferentiate. Its accumulation form says the derivative of g(x) = ∫₀ˣ f(t) dt is f(x); with a variable upper limit u(x), the chain rule gives g′(x) = f(u(x))·u′(x). Integration and differentiation are inverse processes.
FTCAntiderivativeAccumulation Function
Key Concept 3
Antidifferentiation is a toolkit of techniques
To evaluate integrals you build up methods. The basic rules reverse known derivatives (power rule for antiderivatives, ∫(1/x)dx = ln|x| + C, exponentials, and trig). u-substitution reverses the chain rule by letting u be an inner function. Algebraic setups like long division and completing the square prepare an integrand, and BC adds integration by parts, partial fractions, and improper integrals — with Topic 6.14 about choosing the right technique.
Antiderivative Rulesu-SubstitutionBC Techniques
Accumulated change
Total change of a quantity, found by integrating its rate.
Accumulation
Definite integral
∫₀ᵇ f(x) dx; signed area / accumulated change over [a,b].
Integral
Signed area
Area counted positive above and negative below the x-axis.
Integral
Riemann sum
Σ f(xᵢ)Δx approximating area with rectangles.
Approximation
Left / right sum
Rectangle heights from left or right endpoints.
Approximation
Midpoint sum
Rectangle heights from subinterval midpoints.
Approximation
Trapezoidal rule
Approximating area with trapezoids (average of left/right).
Approximation
Antiderivative
F with F′ = f; ∫f dx = F(x) + C.
Antiderivatives
Indefinite integral
∫f(x) dx = F(x) + C; the family of antiderivatives.
Antiderivatives
Constant of integration
The + C capturing all antiderivatives.
Antiderivatives
FTC (evaluation)
∫₀ᵇ f = F(b) − F(a).
FTC
Accumulation function
g(x) = ∫₀ˣ f(t) dt; g′(x) = f(x).
FTC
u-substitution
Reversing the chain rule via u = inner function.
Techniques
Integration by parts (BC)
∫u dv = uv − ∫v du.
Techniques
Improper integral (BC)
An integral with an infinite bound or a discontinuity.