The definite integral and the Newton–Leibniz formula

If f(x) ≥ 0 on [a, b], the figure bounded above by the graph of y = f(x), below by the Ox axis and at the sides by the lines x = a and x = b is called a curvilinear trapezoid. Its area function S(x) satisfies S′(x) = f(x), so S is an antiderivative of f, and the area equals F(b) − F(a) for any antiderivative F of f. This number is the definite integral of f over [a, b]: ∫ₐᵇ f(x) dx = F(b) − F(a) (the Newton–Leibniz formula), written briefly F(x)|ₐᵇ. The definite integral is also the limit of integral sums f(ξₖ)Δxₖ: cut the segment into very thin pieces and add the areas of the rectangles. Properties: ∫ₐᵃ f dx = 0; ∫ₐᵇ f dx = −∫_b^a f dx; ∫ₐᶜ f dx = ∫ₐᵇ f dx + ∫_b^c f dx; if f is even, ∫₋ₐᵃ f dx = 2∫₀ᵃ f dx; if f ≥ 0 the integral is ≥ 0; if f ≤ g then ∫f dx ≤ ∫g dx. The definite integral is a number while the indefinite integral is a family of functions; if f changes sign the integral gives a “signed area” (for example ∫₀^π cos x dx = 0).