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Free jets describe the flow of liquid exiting a reservoir through an opening into the atmosphere without resistance. The velocity (v) of the liquid jet is derived using Bernoulli's principle and expressed as:
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Integration by Parts: Indefinite Integrals01:26

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Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...
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Definite integrals involving the product of two functions over a fixed interval can be evaluated using integration by parts. This method rewrites the integral as the difference of a product evaluated at the endpoints and a remaining definite integral that is often simpler to compute.A representative example is the definite integral of the inverse tangent function. Since there is no direct integration formula for arctan ⁡x, the integrand is rewritten as a product of arctan⁡ x and the...
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Definite Integral01:29

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Consider a real-valued function defined on a closed interval. One of the fundamental objectives in calculus is to determine the area under the graph of such a function. When an exact computation is not readily available, this area can be estimated by dividing the interval into a finite number of equal subintervals. Each subinterval corresponds to a rectangle whose width is the length of the subinterval and whose height is determined by the value of the function at a selected point within that...
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The water inflow rate into a storage tank is not constant but increases over time. Initially, the pump delivers water at a rate of 5 L/min. However, the inflow rate increases by 2 L/min for each additional minute due to rising pressure or system adjustments. This scenario can be described mathematically by a linear function:It is necessary to integrate the inflow rate function to measure the total volume of water added to the tank over time. The total water volume V(t) is obtained by performing...
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Smart speakers process voice commands by modeling audio inputs as piecewise functions and analyzing them through integration against trigonometric functions, such as cosine. This mathematical approach is fundamental in signal processing, where complex sound waves are decomposed into simpler frequency components.Consider a definite integral involving a piecewise function multiplied by a cosine function. Because the function is defined differently over separate intervals, the integral is split...
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Video Experimental Relacionado

Updated: Feb 15, 2026

Planar and Three-Dimensional Printing of Conductive Inks
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Láseres de microdisco integrados en guías de onda mediante impresión por inyección de tinta

Adrian Abazi, Kota Hiramoto, Hayato Matsubayashi

    Optics letters
    |February 13, 2026
    PubMed
    Resumen
    Este resumen es generado por máquina.

    Los investigadores desarrollaron un método novedoso para integrar láseres de bajo costo en circuitos integrados fotónicos utilizando microresonadores de disco impresos por inyección de tinta. Esta técnica permite sistemas de datacom y sensores personalizables para diversas aplicaciones.

    Palabras clave:
    impresión por inyección de tintaláseres de microdiscocircuitos integrados fotónicosacoplamiento verticalresonadores de guía de ondadatacomsensores

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    Área de la Ciencia:

    • Fotónica y Ciencia de Materiales
    • Optoelectrónica y Fotónica Integrada

    Sus antecedentes:

    • Los circuitos integrados fotónicos (PIC) ofrecen soluciones versátiles para la comunicación de datos y el sensado.
    • La integración de fuentes de luz, especialmente láseres, en los PIC es un obstáculo técnico y económico importante.

    Objetivo del estudio:

    • Demostrar un método rentable para acoplar láseres de microdisco impresos por inyección de tinta a circuitos integrados fotónicos apilados verticalmente.
    • Habilitar funcionalidades novedosas y circuitos reconfigurables en una plataforma de chip compacta.

    Principales métodos:

    • Impresión por inyección de tinta de resonadores láser de microdisco dopados con tinte con un bajo umbral de emisión (< 100 µJ/mm²).
    • Acoplamiento vertical de modos de galería de susurros de resonadores en forma de cuña a guías de onda de nitruro de silicio.
    • Alineación precisa tanto en altura como en dimensiones laterales para una transferencia de modo eficiente.

    Principales resultados:

    • Integración exitosa de láseres de microdisco impresos por inyección de tinta en un PIC.
    • Demostración de acoplamiento vertical eficiente entre resonadores y guías de onda.
    • Se alcanzaron bajos umbrales de emisión adecuados para aplicaciones prácticas.

    Conclusiones:

    • La impresión por inyección de tinta ofrece un enfoque flexible y escalable para la fabricación de láseres de microdisco para PIC.
    • Este método facilita el desarrollo de circuitos integrados fotónicos reconfigurables y de bajo costo.
    • La tecnología es prometedora para diversas aplicaciones en datacom, sensado químico y sensado biológico.