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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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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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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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Driving Under the Influence: How Music Listening Affects Driving Behaviors
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Probing sensorimotor integration during musical performance.

Shinichi Furuya1,2, Yuta Furukawa2, Kazumasa Uehara2,3

  • 1Sony Computer Science Laboratories Inc. (Sony CSL), Tokyo, Japan.

Annals of the New York Academy of Sciences
|March 11, 2018
PubMed
Summary
This summary is machine-generated.

New techniques using brain stimulation and force feedback reveal how auditory and proprioceptive senses integrate for skillful piano playing. These methods help understand sensorimotor control in musical performance.

Keywords:
dexterityneuroplasticitypsychophysicssensorimotor integrationtranscranial magnetic stimulation

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Area of Science:

  • Neuroscience
  • Motor Control
  • Music Cognition

Background:

  • Skillful sequential movements in music rely on integrating sensory information (visual, auditory, proprioceptive).
  • Existing research methods struggle to fully explain the online sensorimotor integration crucial for musical performance.
  • Understanding auditory-motor and proprioceptive feedback is key to deciphering complex motor skills.

Purpose of the Study:

  • To introduce and validate novel techniques for investigating sensorimotor integration in piano performance.
  • To explore the specific roles of auditory and proprioceptive feedback in musical skill acquisition and execution.
  • To provide tools for dissecting the neural and computational underpinnings of sensorimotor processes in musicians.

Main Methods:

  • Development of a closed-loop noninvasive brain stimulation system (transcranial magnetic stimulation, motion sensor, microcomputer) to assess auditory-motor integration during piano playing.
  • Implementation of a force-field system to manipulate piano key weight, enabling the study of movement adaptation and internal model formation.
  • Utilizing neurophysiological and psychophysics experiments to validate the efficacy of the developed systems.

Main Results:

  • The proposed techniques effectively assess time-varying cortical processes related to auditory-motor integration.
  • The force-field system successfully characterized movement adaptation, offering insights into the internal representation of the piano.
  • Experimental results validated the systems as powerful tools for studying sensorimotor integration in musical contexts.

Conclusions:

  • The novel techniques provide unprecedented insights into sensorimotor integration during piano performance.
  • These methods are effective for disentangling the complex computational and neural processes involved in musical skill.
  • This research advances our understanding of how sensory feedback shapes motor control in expert musicianship.