integraalarvutuste

Integraalarvutused, commonly referred to as integration in mathematics, is a fundamental concept with wide-ranging applications. It is the process of finding the antiderivative of a function, which is essentially the reverse operation of differentiation. This process allows us to determine the area under a curve, the volume of solids, and to solve differential equations.

The symbol for integration is an elongated S, ∫. An indefinite integral of a function f(x) is denoted as ∫f(x) dx, and it results in a family of functions, F(x) + C, where F'(x) = f(x) and C is the constant of integration. The definite integral, on the other hand, is calculated between two specific limits, say a and b, and is denoted as ∫[a,b] f(x) dx. The result of a definite integral is a single numerical value, representing the net area between the curve of f(x) and the x-axis from x=a to x=b.

The Fundamental Theorem of Calculus establishes the crucial link between differentiation and integration. It states that the definite integral of a function can be evaluated by finding its antiderivative and then subtracting the value of the antiderivative at the lower limit from its value at the upper limit. This theorem revolutionized calculus and made solving many complex problems significantly easier. Applications of integraalarvutused span physics, engineering, economics, statistics, and many other scientific disciplines, making it an indispensable tool in quantitative analysis.