integraalarvutus

Integraalarvutus is the Finnish term for integral calculus, a fundamental branch of mathematics. It is broadly concerned with the study of integration, which is essentially the process of finding the integral of a function. Integration can be understood in two main ways: as finding the antiderivative of a function, or as calculating the area under the curve of a function's graph.

The antiderivative of a function f(x), denoted as F(x), is a function whose derivative is f(x). This is also known as the indefinite integral. The process of finding this antiderivative is called antidifferentiation or indefinite integration. The fundamental theorem of calculus establishes a crucial link between differentiation and integration, stating that differentiation and integration are inverse operations.

The definite integral, on the other hand, is used to calculate the net area between a function's graph and the x-axis over a specified interval. This is achieved by summing up infinitely many infinitesimally small rectangles under the curve. The definite integral has numerous applications in physics, engineering, economics, and other fields, for example, in calculating displacement from velocity, work done by a force, or the volume of solids.

The development of integral calculus is largely attributed to Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, though its roots can be traced back to ancient Greek mathematicians. It is a powerful tool for solving problems involving accumulation, rates of change, and areas.