Integrable

In mathematics, 'integrable' refers to a function or system for which an integral can be found, meaning a function that can be integrated, resulting in a definite numerical answer or another related function. This process essentially reverses differentiation, allowing us to calculate the area under a curve, the volume of a solid, or the total accumulation of a quantity over time. The concept extends beyond basic calculus, applying to more complex differential equations and systems. Specifically, a function is integrable if the integral, typically computed as a Riemann sum, converges to a finite value or a well-defined function.

Integrable meaning with examples

• The physicist carefully tested the new model to determine if the equations of motion were integrable. If so, analytical solutions could be derived, providing deeper insights into the system's behavior and simplifying predictions and simulations, saving time and resources. They were relieved when results showed the system integrable.
• In order to solve the differential equations describing the problem, it was first necessary to establish if the model was integrable. A non-integrable model would demand computationally expensive numerical solutions; whereas, integrable ones often allow for more elegant and insightful analytical expressions, making analysis far easier.
• The engineer sought to streamline the process, by showing the system could be simplified if it was integrable, and could then be solved by finding an equivalent system with the exact same integral. This would reduce computational complexity and make the system more manageable, leading to greater efficiency.
• Mathematicians are always keen to prove if systems are integrable, as this opens doors to a wealth of analytical techniques. The identification of conserved quantities and the ability to determine precise behavior are highly coveted outcomes, yielding powerful understandings of systems.
• The researchers debated if the complex financial model was actually integrable, or if it would require a purely numerical approach. They found the model to be integrable, meaning there were opportunities to improve the efficiency with which they could analyse the system, avoiding costly calculations.