12/2415 Integral calculus is the branch of mathematics concerned with accumulation, the process of adding up infinitely many infinitesimally small quantities to find a whole, and it stands as the inverse operation of differential calculus, which studies rates of change. At its heart lies the integral, a continuous analogue of a sum that allows us to compute areas under curves, volumes of irregular solids, lengths of curved paths, centres of mass, work done by variable forces, and countless other quantities that arise in physics, engineering, economics, and biology. The fundamental idea is to slice a region into very thin pieces of width Δx, approximate the contribution of each slice, and then take the limit as these slices become infinitely narrow (dx), causing the sum to converge to the exact value; this limiting process is formalized through the definite integral ∫ₐᵇ f(x) dx, where the stylized “S” symbol denotes summation, f(x) is the integrand, and a and b are the lower and upper bounds. The power of integral calculus unfolds through the Fundamental Theorem of Calculus, which connects differentiation and integration by showing that if F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a), transforming the daunting task of evaluating limits of Riemann sums into the much simpler problem of finding antiderivatives. While definite integrals yield specific numerical values representing accumulated quantities over an interval, indefinite integrals ∫ f(x) dx produce a family of antiderivative functions F(x) + C, where C is the constant of integration reflecting that any constant vanishes upon differentiation. Mastering integral calculus involves learning techniques such as substitution, integration by parts, partial fractions, and trigonometric substitution, each designed to rewrite complicated integrands into forms we can evaluate, and it opens the door to solving differential equations, modelling dynamic systems, and understanding the accumulated behaviour of functions across continuous domains. 7/2415 we love pu v.2
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