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Related Concept Videos

Integration by Parts: Indefinite Integrals01:26

Integration by Parts: Indefinite Integrals

211
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...
211
Integration by Parts: Definite Integrals01:23

Integration by Parts: Definite Integrals

91
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

Definite Integral

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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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Indefinite Integrals01:25

Indefinite Integrals

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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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Integration by Parts: Problem Solving01:29

Integration by Parts: Problem Solving

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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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Approximate Integration01:24

Approximate Integration

58
In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
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Related Experiment Video

Updated: Feb 9, 2026

Using Micro-Electro-Mechanical Systems MEMS to Develop Diagnostic Tools
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A Fully Integrated Quartz MEMS VHF TCXO.

Randall L Kubena, Frederic P Stratton, Hung D Nguyen

    IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control
    |June 2, 2018
    PubMed
    Summary

    This study presents a fully integrated 32-MHz quartz temperature compensated crystal oscillator (TCXO) using CMOS electronics and wafer-level vacuum packaging. The novel design achieves high frequency stability over temperature with low power consumption.

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    Area of Science:

    • Microelectromechanical Systems (MEMS)
    • Solid-State Electronics
    • Materials Science

    Background:

    • Traditional crystal oscillators face challenges with integration and environmental stability.
    • Achieving high-precision frequency control in miniaturized electronic systems is crucial for advanced applications.

    Purpose of the Study:

    • To develop a fully integrated, wafer-level packaged 32-MHz quartz temperature compensated crystal oscillator (TCXO).
    • To demonstrate stress isolation and precise temperature compensation using novel resonator design and on-chip circuitry.

    Main Methods:

    • Utilized a low-temperature MEMS-after quartz process for wafer-level vacuum packaging.
    • Implemented a novel quartz resonator design for stress isolation from the CMOS substrate.
    • Integrated third-order compensation circuitry for temperature stabilization.

    Main Results:

    • Achieved classical AT-cut frequency/temperature profiles and low hysteresis.
    • Demonstrated temperature compensation to < ±0.2 parts per million over temperature.
    • Operated the TCXO at a low power consumption of 2.5 mW.

    Conclusions:

    • The developed TCXO offers high precision and low power in a fully integrated, wafer-level package.
    • The novel design enables seamless integration with large CMOS wafers using carrier wafer techniques.
    • This technology is suitable for advanced applications requiring stable and miniaturized frequency sources.