Integroimalla

Integroimalla is a Finnish grammatical form meaning “by integrating” or “through integration.” It is the gerund form derived from the verb integroida (to integrate) and the instrumental suffix -malla, which signals the method or means by which something is done. In mathematical writing, integroimalla is used to indicate that a result or step is obtained via the process of integration.

In mathematics, integroimalla describes applying integral calculus to derive a quantity. The phrase is common in explanations and problem statements to emphasize the method rather than merely presenting a numerical answer. It can refer to evaluating indefinite or definite integrals, transforming expressions through integration, or using integral techniques to obtain areas, probabilities, expectations, or other properties described by integrals. While it does not specify a particular technique, it signals that the next step relies on integrating a function with respect to a variable.

A simple example illustrates the idea. If a function has derivative f′(x) = 2x, integroimalla with respect to x yields the antiderivative f(x) = ∫2x dx = x^2 + C. In more applied contexts, integroimalla might be used to derive the area under a curve by integrating the curve’s height over an interval, or to compute expected values in probability theory by integrating a probability density function.

Overall, integroimalla is a concise linguistic marker in Finnish mathematical writing that points to the action of performing integration as part of a solution or demonstration.