integraalilause

Integraalilause, often translated as Integral Theorem or Fundamental Theorem of Calculus, is a cornerstone of calculus that establishes a profound connection between differentiation and integration. It essentially states that integration and differentiation are inverse operations. The theorem exists in two main parts. The first part, often called the differentiation of an integral, says that if you define a function by integrating another function over a variable upper limit, the derivative of this new function will be the original function itself. For instance, if F(x) is defined as the integral of f(t) from a constant 'a' to 'x', then the derivative of F(x) with respect to x is simply f(x). This part is crucial for understanding how to find antiderivatives.

The second part of the Integraalilause, known as the evaluation theorem, provides a powerful method for calculating definite integrals. It states that the definite integral of a function f(x) from a lower limit 'a' to an upper limit 'b' can be found by evaluating the antiderivative of f(x), let's call it F(x), at the upper limit and subtracting its value at the lower limit. Mathematically, this is expressed as the integral of f(x) from a to b equals F(b) - F(a). This theorem dramatically simplifies the calculation of areas under curves and other applications of definite integrals, replacing the cumbersome process of Riemann sums with a straightforward evaluation of an antiderivative. The Integraalilause is thus indispensable for solving a vast array of problems in mathematics, physics, engineering, and economics.