← MATH 01130 · Calculus I
MATH 01130 · Integrals (intro)
Definite integral as area / net change
Definition
∫ₐᵇ f(x) dx is the signed area between y=f(x) and the x-axis from a to b (above positive, below negative). Also equals net change of an antiderivative: F(b)−F(a).
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Riemann sums: sum of f(xᵢ*)Δx approximates the integral; limit of refined partitions is the definition.
Net change theorem: ∫ₐᵇ F′(x) dx = F(b)−F(a) — total accumulation.
Absolute area for regions below the axis requires splitting or integrating |f|.
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1.Interpret ∫₀³ v(t) dt if v is velocity.